The Diffiety Institute Preprint Series

Preprint DIPS–7/98 August 3, 1998

Available via INTERNET: http://ecfor.rssi.ru/˜ diffiety/ http://www.botik.ru/˜ diffiety/ anonymous FTP: ftp://ecfor.rssi.ru/general/pub/diffiety ftp://www.botik.ru/˜ diffiety/preprints

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Homological methods in equations of mathematical physics

by

Joseph KRASIL′SHCHIK and Alexander VERBOVETSKY

math.DG/9808130

HOMOLOGICAL METHODS

IN EQUATIONS OF MATHEMATICAL PHYSICS^{1}

Joseph KRASIL′SHCHIK^{2}

Independent University of Moscow and The Diffiety Institute, Moscow, Russia

and

Alexander VERBOVETSKY^{3}

Moscow State Technical University and The Diffiety Institute, Moscow, Russia

1Lectures given in August 1998 at the International Summer School in Levoča, Slovakia. This work was supported in part by RFBR grant 97-01-00462 and INTAS grant 96-0793

2Correspondence to: J. Krasil′shchik, 1st Tverskoy-Yamskoy per., 14, apt. 45, 125047 Moscow, Russia

E-mail: josephk@glasnet.ru

3Correspondence to: A. Verbovetsky, Profsoyuznaya 98-9-132, 117485 Moscow, Russia

E-mail: verbovet@mail.ecfor.rssi.ru

Contents

Introduction 4 1. Differential calculus over commutative algebras 6 1.1. Linear differential operators 6 1.2. Multiderivations and the Diff-Spencer complex 8 1.3. Jets 11 1.4. Compatibility complex 13 1.5. Differential forms and the de Rham complex 13 1.6. Left and right differential modules 16 1.7. The Spencer cohomology 18 1.8. Geometrical modules 25 2. Algebraic model for Lagrangian formalism 27 2.1. Adjoint operators 27 2.2. Berezinian and integration 28 2.3. Green’s formula 30 2.4. The Euler operator 32 2.5. Conservation laws 34 3. Jets and nonlinear differential equations. Symmetries 35 3.1. Finite jets 35 3.2. Nonlinear differential operators 37 3.3. Infinite jets 39 3.4. Nonlinear equations and their solutions 42 3.5. Cartan distribution on J^{k}(π) 44 3.6. Classical symmetries 49 3.7. Prolongations of differential equations 53 3.8. Basic structures on infinite prolongations 54 3.9. Higher symmetries 62 4. Coverings and nonlocal symmetries 68 4.1. Coverings 68 4.2. Nonlocal symmetries and shadows 71 4.3. Reconstruction theorems 73 5. Frölicher–Nijenhuis brackets and recursion operators 77 5.1. Calculus in form-valued derivations 77 5.2. Algebras with flat connections and cohomology 82 5.3. Applications to differential equations: recursion operators 86 5.4. Passing to nonlocalities 94 6. Horizontal cohomology 99 6.1. C-modules on differential equations 100 6.2. The horizontal de Rham complex 104 6.3. Horizontal compatibility complex 106 6.4. Applications to computing the C-cohomology groups 108 6.5. Example: Evolution equations 109 7. Vinogradov’s C-spectral sequence 111 7.1. Definition of the Vinogradov C-spectral sequence 111 7.2. The term E_{1} for J^{∞}(π) 111 7.3. The term E_{1} for an equation 116 7.4. Example: Abelian p-form theories 118 7.5. Conservation laws and generating functions 120 7.6. Generating functions from the antifield-BRST standpoint 123 7.7. Euler–Lagrange equations 124 7.8. The Hamiltonian formalism on J^{∞}(π) 126 7.9. On superequations 130 Appendix: Homological algebra 132 8.1. Complexes 132 8.2. Spectral sequences 137 References 144

Introduction

Mentioning (co)homology theory in the context of differential equations would sound a bit ridiculous some 30–40 years ago: what could be in common between the essentially analytical, dealing with functional spaces theory of partial differential equations (PDE) and rather abstract and algebraic cohomologies?

Nevertheless, the first meeting of the theories took place in the papers by D. Spencer and his school ([46, 17]), where cohomologies were applied to analysis of overdetermined systems of linear PDE generalizing classical works by Cartan [12]. Homology operators and groups introduced by Spencer (and called the Spencer operators and Spencer homology nowadays) play a basic role in all computations related to modern homological applications to PDE (see below).

Further achievements became possible in the framework of the geometrical approach to PDE. Originating in classical works by Lie, Bäcklund, Darboux, this approach was developed by A. Vinogradov and his co-workers (see [32, 61]). Treating a differential equation as a submanifold in a suitable jet bundle and using a nontrivial geometrical structure of the latter allows one to apply powerful tools of modern differential geometry to analysis of nonlinear PDE of a general nature. And not only this: speaking the geometrical language makes it possible to clarify underlying algebraic structures, the latter giving better and deeper understanding of the whole picture, [32, Ch. 1] and [58, 26].

It was also A. Vinogradov to whom the next homological application to PDE belongs. In fact, it was even more than an application: in a series of papers [59, 60, 63], he has demonstrated that the adequate language for Lagrangian formalism is a special spectral sequence (the so-called Vinogradov C-spectral sequence) and obtained first spectacular results using this language. As it happened, the area of the C-spectral sequence applications is much wider and extends to scalar differential invariants of geometric structures [57], modern field theory [5, 6, 3, 9, 18], etc. A lot of work was also done to specify and generalize Vinogradov’s initial results, and here one could mention those by I. M. Anderson [1, 2], R. L. Bryant and P. A. Griffiths [11], D. M. Gessler [16, 15], M. Marvan [39, 40], T. Tsujishita [47, 48, 49], W. M. Tulczyjew [50, 51, 52].

Later, one of the authors found out that another cohomology theory (Ccohomologies) is naturally related to any PDE [24]. The construction uses the fact that the infinite prolongation of any equation is naturally endowed with a flat connection (the Cartan connection). To such a connection, one puts into correspondence a differential complex based on the Frölicher– Nijenhuis bracket [42, 13]. The group H^{0} for this complex coincides with the symmetry algebra of the equation at hand, the group H^{1} consists of equivalence classes of deformations of the equation structure. Deformations of a special type are identified with recursion operators [43] for symmetries. On the other hand, this theory seems to be dual to the term E_{1} of the Vinogradov C-spectral sequence, while special cochain maps relating the former to the latter are Poisson structures on the equation [25].

Not long ago, the second author noticed ([56]) that both theories may be understood as horizontal cohomologies with suitable coefficients. Using this observation combined with the fact that the horizontal de Rham cohomology is equal to the cohomology of the compatibility complex for the universal linearization operator, he found a simple proof of the vanishing theorem for the term E_{1} (the “k-line theorem”) and gave a complete description of C-cohomology in the “2-line situation”.

Our short review will not be complete, if we do not mention applications of cohomologies to the singularity theory of solutions of nonlinear PDE ([35]), though this topics is far beyond the scope of these lecture notes.

⋆ ⋆ ⋆

The idea to expose the above mentioned material in a lecture course at the Summer School in Levoča belongs to Prof. D. Krupka to whom we are extremely grateful.

We tried to give here a complete and self-contained picture which was not easy under natural time and volume limitations. To make reading easier, we included the Appendix containing basic facts and definitions from homological algebra. In fact, the material needs not 5 days, but 3–4 semester course at the university level, and we really do hope that these lecture notes will help to those who became interested during the lectures. For further details (in the geometry of PDE especially) we refer the reader to the books [32] and [34] (an English translation of the latter is to be published by the American Mathematical Society in 1999). For advanced reading we also strongly recommend the collection [19], where one will find a lot of cohomological applications to modern physics.

J. Krasil′shchik A. Verbovetsky Moscow, 1998

1. Differential calculus over commutative algebras

Throughout this section we shall deal with a commutative algebra A over a field 𝕜 of zero characteristic. For further details we refer the reader to [32, Ch. I] and [26].

1.1. Linear differential operators. Consider two A-modules P and Q and the group Hom_{𝕜}(P, Q). Two A-module structures can be introduced into this group:

(aΔ)(p) = aΔ(p), (a^{+}Δ)(p) = Δ(ap),  (1.1)

where a ∈ A, p ∈ P, Δ ∈ Hom_{𝕜}(P, Q). We also set

δ_{a}(Δ) = a^{+}Δ − aΔ, δ_{a_{0},...,a_{k}} = δ_{a_{0}} ∘ ⋯ ∘ δ_{a_{k}},

a_{0}, …, a_{k} ∈ A. Obviously, δ_{a,b} = δ_{b,a} and δ_{ab} = aδ_{b} + bδ_{a} for any a, b ∈ A.

Definition 1.1. A 𝕜-homomorphism Δ: P → Q is called a linear differential operator of order ≤ k over the algebra A, if δ_{a_{0},...,a_{k}}(Δ) = 0 for all a_{0}, …, a_{k} ∈ A.

Proposition 1.1. If M is a smooth manifold, ξ, ζ are smooth locally trivial vector bundles over M, A = C^{∞}(M) and P = Γ(ξ), Q = Γ(ζ) are the modules of smooth sections, then any linear differential operator acting from ξ to ζ is an operator in the sense of Definition 1.1 and vice versa.

Exercise 1.1. Prove this fact.

Obviously, the set of all differential operators of order ≤ k acting from P to Q is a subgroup in Hom_{𝕜}(P, Q) closed with respect to both multiplications (1.1). Thus we obtain two modules denoted by Diff_{k}(P, Q) and Diff_{k}^{+}(P, Q) respectively. Since a(b^{+}Δ) = b^{+}(aΔ) for any a, b ∈ A and Δ ∈ Hom_{𝕜}(P, Q), this group also carries the structure of an A-bimodule denoted by Diff_{k}^{(+)}(P, Q). Evidently, Diff_{0}(P, Q) = Diff_{0}^{+}(P, Q) = Hom_{A}(P, Q).

It follows from Definition 1.1 that any differential operator of order ≤ k is an operator of order ≤ l for all l ≥ k and consequently we obtain the embeddings Diff_{k}^{(+)}(P, Q) ⊂ Diff_{l}^{(+)}(P, Q), which allow us to define the filtered bimodule Diff^{(+)}(P, Q) = ⋃_{k≥0} Diff_{k}^{(+)}(P, Q).

We can also consider the ℤ-graded module associated to the filtered module Diff^{(+)}(P, Q): Smbl(P, Q) = ⨁_{k≥0} Smbl_{k}(P, Q), where Smbl_{k}(P, Q) = Diff_{k}^{(+)}(P, Q)/Diff_{k−1}^{(+)}(P, Q), which is called the module of symbols. The elements of Smbl(P, Q) are called symbols of operators acting from P to Q. It easily seen that two module structures defined by (1.1) become identical in Smbl(P, Q).

The following properties of linear differential operator are directly implied by the definition:

Proposition 1.2. Let P, Q and R be A-modules. Then:

(1) If Δ_{1} ∈ Diff_{k}(P, Q) and Δ_{2} ∈ Diff_{l}(Q, R) are two differential operators, then their composition Δ_{2} ∘ Δ_{1} lies in Diff_{k+l}(P, R).

(2) The maps

i^{·,+} : Diff_{k}(P, Q) → Diff_{k}^{+}(P, Q), i^{+,·}: Diff_{k}^{+}(P, Q) → Diff_{k}(P, Q)

generated by the identical map of Hom_{𝕜}(P, Q) are differential operators of order ≤ k.

Corollary 1.3. There exists an isomorphism

Diff^{+}(P, Diff^{+}(Q, R)) = Diff^{+}(P, Diff(Q, R))

generated by the operators i^{·,+} and i^{+,·}.

Introduce the notation Diff_{k}^{(+)}(Q) = Diff_{k}^{(+)}(A, Q) and define the map D_{k}: Diff_{k}^{+}(Q) → Q by setting D_{k}(Δ) = Δ(1). Obviously, D_{k} is an operator of order ≤ k. Let also

ψ: Diff_{k}^{+}(P, Q) → Hom_{A}(P, Diff_{k}^{+}(Q)), Δ ↦ ψ_{Δ},  (1.2)

be the map defined by (ψ_{Δ}(p))(a) = Δ(ap), p ∈ P, a ∈ A.

Proposition 1.4. The map (1.2) is an isomorphism of A-modules.

Proof. Compatibility of ψ with A-module structures is obvious. To complete the proof it suffices to note that the correspondence

Hom_{A}(P, Diff_{k}^{+}(Q)) ∋ φ ↦ D_{k} ∘ φ ∈ Diff_{k}^{+}(P, Q)

is inverse to ψ.

The homomorphism ψ_{Δ} is called Diff-associated to Δ.

Remark 1.1. Consider the correspondence P ⇒ Diff_{k}^{+}(P, Q) and for any A-homomorphism f: P → R define the homomorphism

Diff_{k}^{+}(f, Q): Diff_{k}^{+}(R, Q) → Diff_{k}^{+}(P, Q)

by setting Diff_{k}^{+}(f, Q)(Δ) = Δ ∘ f. Thus, Diff_{k}^{+}(·, Q) is a contravariant functor from the category of all A-modules to itself. Proposition 1.4 means that this functor is representable and the module Diff_{k}^{+}(Q) is its representative object. Obviously, the same is valid for the functor Diff^{+}(·, Q) and the module Diff^{+}(Q).

From Proposition 1.4 we also obtain the following

Corollary 1.5. There exists a unique homomorphism

c_{k,l} = c_{k,l}(P): Diff_{k}^{+}(Diff_{l}^{+}(P)) → Diff_{k+l}(P)

such that the diagram

Diff_{k}^{+}(Diff_{l}^{+}(P)) ⟶^{D_{k}} Diff_{l}^{+}(P)
c_{k,l}↓  ↓^{D_{l}}
Diff_{k+l}^{+}(P) ⟶^{D_{k+l}}  P

is commutative.

Proof. It suffices to use the fact that the composition

D_{l} ∘ D_{k} : Diff_{k}(Diff_{l}(P)) ⟶ P

is an operator of order ≤ k + l and to set c_{k,l} = ψ_{D_{l}∘D_{k}}.

The map c_{k,l} is called the gluing homomorphism and from the definition it follows that (c_{k,l}(Δ))(a) = (Δ(a))(1), Δ ∈ Diff_{k}^{+}(Diff_{l}^{+}(P)), a ∈ A.

Remark 1.2. The correspondence P ⇒ Diff_{k}^{+}(P) also becomes a (covariant) functor, if for a homomorphism f: P → Q we define the homomorphism Diff_{k}^{+}(f): Diff_{k}^{+}(P) → Diff_{k}^{+}(Q) by Diff_{k}^{+}(f)(Δ) = f ∘ Δ. Then the correspondence P ⇒ c_{k,l}(P) is a natural transformation of functors Diff_{k}^{+}(Diff_{l}^{+}(·)) and Diff_{k+l}^{+}(·) which means that for any A-homomorphism f: P → Q the diagram

Diff_{k}^{+}(Diff_{l}^{+}(P))

Diff_{k}^{+}(Diff_{l}^{+}(f))
⟶ Diff_{k}^{+}(Diff_{l}^{+}(Q))
c_{k,l}(P)↓  ↓^{c_{k,l}(Q)}
Diff_{k+l}^{+}(P)  ⟶^{Diff_{k+l}^{+}(f)}  Diff_{k+l}^{+}(Q)

is commutative.

Note also that the maps c_{k,l} are compatible with the natural embeddings Diff_{k}^{+}(P) → Diff_{s}^{+}(P), k ≤ s, and thus we can define the gluing c_{∗,∗}: Diff^{+}(Diff^{+}(·)) → Diff^{+}(·).

1.2. Multiderivations and the Diff-Spencer complex. Let A^{⊗k} = A ⊗_{𝕜} ⋯ ⊗_{𝕜} A, k times.

Definition 1.2. A 𝕜-linear map ∇: A^{⊗k} → P is called a skew-symmetric multiderivation of A with values in an A-module P, if the following conditions hold:

(1) ∇(a_{1}, …, a_{i}, a_{i+1}, …, a_{k}) + ∇(a_{1}, …, a_{i+1}, a_{i}, …, a_{k}) = 0,
(2) ∇(a_{1}, …, a_{i−1}, ab, a_{i+1}, …, a_{k}) =

a∇(a_{1}, …, a_{i−1}, b, a_{i+1}, …, a_{k}) + b∇(a_{1}, …, a_{i−1}, a, a_{i+1}, …, a_{k})

for all a, b, a_{1}, …, a_{k} ∈ A and any i, 1 ≤ i ≤ k.

The set of all skew-symmetric k-derivations forms an A-module denoted by D_{k}(P). By definition, D_{0}(P) = P. In particular, elements of D_{1}(P) are called P -valued derivations and form a submodule in Diff_{1}(P) (but not in the module Diff_{1}^{+}(P)!).

There is another, functorial definition of the modules D_{k}(P): for any ∇ ∈ D_{k}(P) and a ∈ A we set (a∇)(a_{1}, …, a_{k}) = a∇(a_{1}, …, a_{k}). Note first that the composition γ_{1} : D_{1}(P) ↪ Diff_{1}(P) ⟶^{i^{·,+}} Diff_{1}^{+}(P) is a monomorphic differential operator of order ≤ 1. Assume now that the first-order monomorphic operators γ_{i} = γ_{i}(P): D_{i}(P) → D_{i−1}(Diff_{1}^{+}(P)) were defined for all i ≤ k. Assume also that all the maps γ_{i} are natural^{4} operators. Consider the composition

D_{k}(Diff_{1}^{+}(P)) ⟶^{γ_{k}} D_{k−1}(Diff_{1}^{+}(Diff_{1}^{+}(P))) ⟶^{D_{k−1}(c_{1,1})} D_{k−1}(Diff_{2}^{+}(P)). (1.3)

Proposition 1.6. The following facts are valid:

(1) D_{k+1}(P) coincides with the kernel of the composition (1.3).

(2) The embedding γ_{k+1} : D_{k+1}(P) ↪ D_{k}(Diff_{1}^{+}(P)) is a first-order differential operator.

(3) The operator γ_{k+1} is natural.

The proof reduces to checking the definitions.

Remark 1.3. We saw above that the A-module D_{k+1}(P) is the kernel of the map D_{k−1}(c_{1,1})∘γ_{k}, the latter being not an A-module homomorphism but a differential operator. Such an effect arises in the following general situation.

Let F be a functor acting on a subcategory of the category of A-modules. We say that F is 𝕜-linear, if the corresponding map F_{P,Q}: Hom_{𝕜}(P, Q) → Hom_{𝕜}(P, Q) is linear over 𝕜 for all P and Q from our subcategory. Then we can introduce a new A-module structure in the the 𝕜-module F(P) by setting a q̇ = (F(a))(q), where q ∈ F(P) and F(a): F(P) → F(P) is the homomorphism corresponding to the multiplication by a: p ↦ ap, p ∈ P. Denote the module arising in such a way by F˙ (P).

Consider two 𝕜-linear functors F and G and a natural transformation Δ: P ⇒ Δ(P) ∈ Hom_{𝕜}(F(P), G(P)).

Exercise 1.2. Prove that the natural transformation Δ induces a natural homomorphism of A-modules Δ˙ : F˙ (P) → G˙ (P) and thus its kernel is always an A-module.

From Definition 1.2 on the preceding page it also follows that elements of the modules D_{k}(P), k ≥ 2, may be understood as derivations Δ: A → D_{k−1}(P) satisfying (Δ(a))(b) = −(Δ(b))(a). We call Δ(a) the evaluation of the multiderivation Δ at the element a ∈ A. Using this interpretation, define by induction on k + l the operation ∧: D_{k}(A) ⊗_{A} D_{l}(P) → D_{k+l}(P) by setting

4This means that for any A-homomorphism f: P → Q one has γ_{i}(Q) ∘ D_{i}(f) = D_{i−1}(Diff_{1}^{+}(f)) ∘ γ_{i}(P).

a ∧ p = ap, a ∈ D_{0}(A) = A, p ∈ D_{0}(P) = P,

and

(Δ ∧ ∇)(a) = Δ ∧ ∇(a) + (−1)^{l}Δ(a) ∧ ∇.  (1.4)

Using elementary induction on k + l, one can easily prove the following

Proposition 1.7. The operation ∧ is well defined and satisfies the following properties:

(1) Δ ∧ (Δ′ ∧ ∇) = (Δ ∧ Δ′) ∧ ∇,
(2) (aΔ + a′Δ′) ∧ ∇ = aΔ ∧ ∇ + a′Δ′ ∧ ∇,
(3) Δ ∧ (a∇ + a′∇′) = aΔ ∧ ∇ + a′Δ ∧ ∇′,
(4) Δ ∧ Δ′ = (−1)^{kk′}Δ′ ∧ Δ

for any elements a, a′ ∈ A and multiderivations Δ ∈ D_{k}(A), Δ′ ∈ D_{k′}(A), ∇ ∈ D_{l}(P), ∇′ ∈ D_{l′}(P).

Thus, D_{∗}(A) = ⨁_{k≥0} D_{k}(A) becomes a ℤ-graded commutative algebra and D_{∗}(P) = ⨁_{k≥0} D_{k}(P) is a graded D_{∗}(A)-module. The correspondence P ⇒ D_{∗}(P) is a functor from the category of A-modules to the category of graded D_{∗}(A)-modules.

Let now ∇ ∈ D_{k}(Diff_{l}^{+}(P)) be a multiderivation. Define

(S(∇)(a_{1}, …, a_{k−1}))(a) = (∇(a_{1}, …, a_{k−1}, a)(1)),  (1.5)

a, a_{1}, …, a_{k−1} ∈ A. Thus we obtain the map

S: D_{k}(Diff_{l}^{+}(P)) → D_{k−1}(Diff_{l+1}^{+}(P))

which can be represented as the composition

D_{k}(Diff_{l}^{+}(P)) ⟶^{γ_{k}} D_{k−1}(Diff_{1}^{+}(Diff_{l}^{+}(P))) ⟶^{D_{k−1}(c_{1,l})} D_{k−1}(Diff_{l+1}^{+}(P)).

(1.6)

Proposition 1.8. The maps S: D_{k}(Diff_{l}^{+}(P)) → D_{k−1}(Diff_{l+1}^{+}(P)) possess the following properties:

(1) S is a differential operator of order ≤ 1.

(2) S ∘ S = 0.

Proof. The first statement follows from (1.6), the second one is implied by (1.5).

Definition 1.3. The operator S is called the Diff-Spencer operator. The sequence of operators

0 ⟵ P ⟵^{D} Diff^{+}(P) ⟵^{S} Diff^{+}(P) ⟵^{S} D_{2}(Diff^{+}(P)) ⟵ ⋯

is called the Diff-Spencer complex.

1.3. Jets. Now we shall deal with the functors Q ⇒ Diff_{k}(P, Q) and their representability.

Consider an A-module P and the tensor product A ⊗_{𝕜} P. Introduce an A-module structure in this tensor product by setting

a(b ⊗ p) = (ab) ⊗ p, a, b ∈ A, p ∈ P,

and consider the 𝕜-linear map ϵ: P → A ⊗_{𝕜} P defined by ϵ(p) = 1 ⊗ p. Denote by μ^{k} the submodule in A ⊗_{𝕜} P generated by the elements of the form (δ_{a_{0},...,a_{k}}(ϵ))(p) for all a_{0}, …, a_{k} ∈ A and p ∈ P.

Definition 1.4. The quotient module (A ⊗_{𝕜} P)/μ^{k} is called the module of k-jets for P and is denoted by J^{k}(P).

We also define the map j_{k}: P → J^{k}(P) by setting j_{k}(p) = ϵ(p) mod μ^{k}. Directly from the definition of μ^{k} it follows that j_{k} is a differential operator of order ≤ k.

Proposition 1.9. There exists a canonical isomorphism

ψ: Diff_{k}(P, Q) → Hom_{A}(J^{k}(P), Q), Δ ↦ ψ^{Δ},  (1.7)

defined by the equality Δ = ψ^{Δ} ∘ j_{k} and called Jet-associated to Δ.

Proof. Note first that since the module J^{k}(P) is generated by the elements of the form j_{k}(p), p ∈ P, the homomorphism ψ^{Δ}, if defined, is unique. To establish existence of ψ^{Δ}, consider the homomorphism

η: Hom_{A}(A ⊗_{𝕜} P, Q) → Hom_{𝕜}(P, Q), η(φ) = φ ∘ ϵ.

Since φ is an A-homomorphism, one has

δ_{a}(η(φ)) = δ_{a}(φ ∘ ϵ) = φ ∘ δ_{a}(ϵ) = η(δ_{a}(φ)), a ∈ A.

Consequently, the element η(φ) is an operator of order ≤ k if and only if φ(μ^{k}) = 0, i.e., restricting η to Diff_{k}(P, Q) ⊂ Hom_{𝕜}(P, Q) we obtain the desired isomorphism ψ.

The proposition proved means that the functor Q ⇒ Diff_{k}(P, Q) is representable and the module J^{k}(P) is its representative object.

Note that the correspondence P ⇒ J^{k}(P) is a functor itself: if φ: P → Q is an A-module homomorphism, we are able to define the homomorphism J^{k}(φ): J^{k}(P) → J^{k}(Q) by the commutativity condition

P ⟶^{j_{k}} J^{k}(P)
φ↓  ↓^{J^{k}(φ)}
Q ⟶^{j_{k}} J^{k}(Q)

The universal property of the operator j_{k} allows us to introduce the natural transformation c^{k,l} of the functors J^{k+l}(·) and J^{k}(J^{l}(·)) defined by the commutative diagram

P  ⟶^{j_{l}} J^{l}(P)
j_{k+l}↓  ↓^{j_{k}}
J^{k+l}(P) ⟶^{c^{k,l}} J^{k}(J^{l}(P))

It is called the co-gluing homomorphism and is dual to the gluing one discussed in Remark 1.2 on page 8.

Another natural transformation related to functors J^{k}(·) arises from the embeddings μ^{l} ↪ μ^{k}, l ≥ k, which generate the projections ν_{l,k}: J^{l}(P) → J^{k}(P) dual to the embeddings Diff_{k}(P, Q) ↪ Diff_{l}(P, Q). One can easily see that if f: P → P′ is an A-module homomorphism, then J^{k}(f) ∘ ν_{l,k} = ν_{l,k} ∘ J^{l}(f). Thus we obtain the sequence of projections

⋯ ⟶ J^{k}(P) ⟶^{ν_{k,k−1}} J^{k−1}(P) ⟶ ⋯ ⟶ J^{1}(P) ⟶^{ν_{1,0}} J^{0}(P) = P

and set J^{∞}(P) = proj lim J^{k}(P). Since ν_{l,k} ∘ j_{l} = j_{k}, we can also set j_{∞} = proj lim j_{k}: P → J^{∞}(P).

Let Δ: P → Q be an operator of order ≤ k. Then for any l ≥ 0 we have the commutative diagram

P  ⟶^{Δ} Q
j_{k+l}↓  ↓^{j_{l}}
J^{k+l}(P) ⟶^{ψ_{l}^{Δ}} J^{l}(Q)

where ψ_{l}^{Δ} = ψ^{j_{l}∘Δ}. Moreover, if l′ ≥ l, then ν_{l′,l} ∘ ψ_{l′}^{Δ} = ψ_{l}^{Δ} ∘ ν_{k+l′,k+l} and we obtain the homomorphism ψ_{∞}^{Δ} : J^{∞}(P) → J^{∞}(Q).

Note that the co-gluing homomorphism is a particular case of the above construction: c^{k,l} = ψ_{k}^{j_{l}}. Thus, passing to the inverse limits, we obtain the co-gluing c^{∞,∞}:

P ⟶^{j_{∞}} J^{∞}(P)
j_{∞}↓  ↓^{j_{∞}}
J^{∞}(P) ⟶^{c^{∞,∞}} J^{∞}(J^{∞}(P))

1.4. Compatibility complex. The following construction will play an important role below.

Consider a differential operator Δ: Q → Q_{1} of order ≤ k. Without loss of generality we may assume that its Jet-associated homomorphism ψ^{Δ} : J^{k}(Q) → Q_{1} is epimorphic. Choose an integer k_{1} ≥ 0 and define Q_{2} as the cokernel of the homomorphism ψ_{k_{1}}^{Δ} : J^{k+k_{1}}(Q) → J^{k}(Q_{1}),

0 → J^{k+k_{1}}(Q) ⟶^{ψ_{k_{1}}^{Δ}} J^{k_{1}}(Q_{1}) → Q_{2} → 0.

Denote the composition of the operator j_{k_{1}} : Q_{1} → J^{k_{1}}(Q_{1}) with the natural projection J^{k_{1}}(Q_{1}) → Q_{2} by Δ_{1} : Q_{1} → Q_{2}. By construction, we have

Δ_{1} ∘ Δ = ψ^{Δ_{1}} ∘ j_{k_{1}} ∘ Δ = ψ^{Δ_{1}} ∘ ψ_{k_{1}}^{Δ} ∘ j_{k+k_{1}}.

Exercise 1.3. Prove that Δ_{1} is a compatibility operator for the operator Δ, i.e., for any operator ∇ such that ∇ ∘ Δ = 0 and ord ∇ ≥ k_{1}, there exists an operator □ such that ∇ = □ ∘ Δ_{1}.

We can now apply the procedure to the operator Δ_{1} and some integer k_{2} obtaining Δ_{2} : Q_{2} → Q_{3}, etc. Eventually, we obtain the complex

0 ⟶ Q ⟶^{Δ} Q_{1} ⟶^{Δ_{1}} Q_{2} ⟶^{Δ_{2}} ⋯ ⟶ Q_{i} ⟶^{Δ_{i}} Q_{i+1} ⟶ ⋯

which is called the compatibility complex of the operator Δ.

1.5. Differential forms and the de Rham complex. Consider the embedding β: A → J^{1}(A) defined by β(a) = aj_{1}(1) and define the module Λ^{1} = J^{1}(A)/ im β. Let d be the composition of j_{1} and the natural projection J^{1}(A) → Λ^{1}. Then d: A → Λ^{1} is a differential operator of order ≤ 1 (and, moreover, lies in D_{1}(Λ^{1})).

Let us now apply the construction of the previous subsection to the operator d setting all k_{i} equal to 1 and preserving the notation d for the operators d_{i}. Then we get the compatibility complex

0 ⟶ A ⟶^{d} Λ^{1} ⟶^{d} Λ^{2} ⟶ ⋯ ⟶ Λ^{k} ⟶^{d} Λ^{k+1} ⟶ ⋯

which is called the de Rham complex of the algebra A. The elements of Λ^{k} are called k-forms over A.

Proposition 1.10. For any k ≥ 0, the module Λ^{k} is the representative object for the functor D_{k}(·).

Proof. It suffices to compare the definition of Λ^{k} with the description of D_{k}(P) given by Proposition 1.6 on page 9.

Remark 1.4. In the case k = 1, the isomorphism between Hom_{A}(Λ^{1}, ·) and D_{1}(·) can be described more exactly. Namely, from the definition of the operator d: A → Λ^{1} and from Proposition 1.9 on page 11 it follows that any derivation ∇: A → P is uniquely represented as the composition ∇ = φ^{∇}∘d for some homomorphism φ^{∇}: Λ^{1} → P.

As a consequence Proposition 1.10 on the page before, we obtain the following

Corollary 1.11. The module Λ^{k} is the k-th exterior power of Λ^{1}.

Exercise 1.4. Since D_{k}(P) = Hom_{A}(Λ^{k}, P), one can introduce the pairing ⟨·, ·⟩: D_{k}(P) ⊗ Λ^{k} ⟶ P. Prove that the evaluation operation (see p. 9) and the wedge product are mutually dual with respect to this pairing, i.e.,

⟨X, da ∧ ω⟩ = ⟨X(a), ω⟩

for all X ∈ D_{k+1}(P), ω ∈ Λ^{k}, and a ∈ A.

The following proposition establishes the relation of the de Rham differential to the wedge product.

Proposition 1.12 (the Leibniz rule). For any ω ∈ Λ^{k} and θ ∈ Λ^{l} one has

d(ω ∧ θ) = dω ∧ θ + (−1)^{k}ω ∧ dθ.

Proof. We first consider the case l = 0, i.e., θ = a ∈ A. To do it, note that the wedge product ∧: Λ^{k} ⊗_{A} Λ^{l} → Λ^{k+l}, due to Proposition 1.10 on the preceding page, induces the natural embeddings of modules D_{k+l}(P) → D_{k}(D_{l}(P)). In particular, the embedding D_{k+1}(P) → D_{k}(D_{1}(P)) can be represented as the composition

D_{k+l}(P) ⟶^{γ_{k+1}} D_{k}(Diff_{1}^{+}(P)) ⟶^{λ} D_{k}(D_{1}(P)),

where (λ(∇))(a_{1}, …, a_{k}) = ∇(a_{1}, …, a_{k}) − (∇(a_{1}, …, a_{k}))(1). In a dual way, the wedge product is represented as

Λ^{k} ⊗_{A} Λ^{1} ⟶^{λ′} J^{1}(Λ^{k}) ⟶^{ψ^{d}} Λ^{k+1},

where λ′(ω ⊗ da) = (−1)^{k}(j_{1}(ωa) − j_{1}(ω)a). Then

(−1)^{k} ∧ ωda = (−1)^{k}ψ^{d}(λ′(ω ⊗ da))

= ψ^{d}(j_{1}(ωa) − j_{1}(ω)a) = d(ωa) − d(ω)a.

The general case is implied by the identity

d(ω ∧ da) = (−1)^{k}d(d(ωa) − dω · a) = (−1)^{k+1}d(dω · a).

Let us return back to Proposition 1.10 on the page before and consider the A-bilinear pairing

⟨·, ·⟩: D_{k}(P) ⊗_{A} Λ^{k} → P

again. Take a form ω ∈ Λ^{k} and a derivation X ∈ D_{1}(A). Using the definition of the wedge product in D_{∗}(P) (see equality (1.4) on page 10), we can set

⟨Δ, i_{X}ω⟩ = (−1)^{k−1}⟨X ∧ Δ, ω⟩  (1.8)

for an arbitrary Δ ∈ D_{k−1}(P).

Definition 1.5. The operation i_{X}: Λ^{k} → Λ^{k−1} defined by (1.8) on the preceding page is called the internal product, or contraction.

Proposition 1.13. For any X, Y ∈ D_{1}(A) and ω ∈ Λ^{k}, θ ∈ Λ^{l} one has

(1) i_{X}(ω ∧ θ) = i_{X}(ω) ∧ θ + (−1)^{k}ω ∧ i_{X}(θ),
(2) i_{X} ∘ i_{Y} = −i_{Y} ∘ i_{X}

In other words, internal product is a derivation of the ℤ-graded algebra Λ^{∗} = ⨁_{k≥0} Λ^{k} of degree −1 and i_{X}, i_{Y} commute as graded maps.

Consider a derivation X ∈ D_{1}(A) and set

L_{X}(ω) = [i_{X}, d](ω) = i_{X}(d(ω)) + d(i_{X}(ω)), ω ∈ Λ^{∗}.  (1.9)

Definition 1.6. The operation L_{X}: Λ^{∗} → Λ^{∗} defined by 1.9 is called the Lie derivative.

Directly from the definition one obtains the following properties of Lie derivatives:

Proposition 1.14. Let X, Y ∈ D_{1}(A), ω, θ ∈ Λ^{∗}, a ∈ A, α, β ∈ 𝕜. Then the following identities are valid:

(1) L_{αX+βY} = αL_{X} + βL_{Y} ,
(2) L_{aX} = aL_{X} + da ∧ i_{X},
(3) L_{X}(ω ∧ θ) = L_{X}(ω) ∧ θ + ω ∧ L_{X}(θ),
(4) [d, L_{X}] = d ∘ L_{X} − L_{X} ∘ d = 0,
(5) L_{[X,Y]} = [L_{X}, L_{Y}], where [X, Y] = X ∘ Y − Y ∘ X,
(6) i_{[X,Y]} = [L_{X}, i_{Y}] = [i_{X}, L_{Y}].

To conclude this subsection, we present another description of the Diff-Spencer complex. Recall Remark 1.3 on page 9 and introduce the “dotted” structure into the modules D_{k}(Diff_{l}^{+}(P)) and note that Diff_{l}^{+}(P)˙ = Diff_{l}(P). Define the isomorphism

ζ: (D_{k}(Diff^{+}))˙ (P) = Hom_{A}(Λ^{k}, Diff^{+})˙ = Diff^{+}(Λ^{k}, P)˙ = Diff(Λ^{k}, P).

Then we have

Proposition 1.15. The above defined map ζ generates the isomorphism of complexes

⋯ ⟵ (D_{k−1}(Diff^{+}))˙ (P) ⟵^{S}˙ (D_{k}(Diff^{+}))˙ (P) ⟵ ⋯
ζ↓  ^{ζ}↓
⋯ ⟵ Diff(Λ^{k−1}, P) ⟵^{v} Diff(Λ^{k}, P) ⟵ ⋯

where S˙ is the operator induced on “dotted” modules by the Diff-Spencer operator, while v(∇) = ∇ ∘ d.

1.6. Left and right differential modules. From now on till the end of this section we shall assume the modules under consideration to be projective.

Definition 1.7. An A-module P is called a left differential module, if there exists an A-module homomorphism λ: P → J^{∞}(P) satisfying ν_{∞,0}∘λ = id_{P} and such that the diagram

P ⟶^{λ} J^{∞}(P)
λ↓  ↓^{J^{∞}(λ)}
J^{∞}(P) ⟶^{c^{∞,∞}} J^{∞}(J^{∞}(P))

is commutative.

Lemma 1.16. Let P be a left differential module. Then for any differential operator Δ: Q_{1} → Q_{2} there exists an operator Δ_{P} : Q_{1} ⊗_{A} P → Q_{2} ⊗_{A} P satisfying (id_{Q})_{P} = id_{Q⊗_{A}P} for Q = Q_{1} = Q_{2} and

(Δ_{2} ∘ Δ_{1})_{P} = (Δ_{2})_{P} ∘ (Δ_{1})_{P}

for any operators Δ_{1} : Q_{1} → Q_{2}, Δ_{2} : Q_{2} → Q_{3}.

Proof. Consider the map

\overline{Δ}: Q_{1} ⊗_{A} (A ⊗_{𝕜} P) → Q_{2} ⊗_{A} P, q ⊗ a ⊗ p ↦ Δ(aq) ⊗ p.

Since

\overline{Δ}(q ⊗ δ_{a}(ϵ)(p)) = \overline{δ_{a}Δ}(q ⊗ 1 ⊗ p), p ∈ P, q ∈ Q_{1}, a ∈ A,

the map

ξ_{P} (Δ): Q_{1} ⊗_{A} J^{∞}(P) → Q_{2} ⊗_{A} P

is well defined. Set now the operator Δ_{P} to be the composition

Q_{1} ⊗_{A} P ⟶^{id⊗λ} Q_{1} ⊗_{A} J^{∞}(P) ⟶^{ξ_{P} (Δ)} Q_{2} ⊗_{A} P,

which is a k-th order differential operator in an obvious way. Evidently, (id_{Q})_{P} = id_{Q⊗_{A}P}.

Now,

(Δ_{2} ∘ Δ_{1})_{P} = ξ_{P} (Δ_{2} ∘ Δ_{1}) ∘ (id ⊗ λ)
= ξ_{P} (Δ_{2}) ∘ ξ_{J^{∞}(P)}(Δ_{1}) ∘ (id ⊗ c^{∞,∞}) ∘ (id ⊗ λ)
= ξ_{P} (Δ_{2}) ∘ ξ_{J^{∞}(P)}(Δ_{1}) ∘ (id ⊗ J^{∞}(λ)) ∘ (id ∘ λ)
= ξ_{P} (Δ_{2}) ∘ (id ⊗ λ) ∘ ξ_{P} (Δ_{1}) ∘ (id ⊗ λ) = (Δ_{2})_{P} ∘ (Δ_{1})_{P} ,

which proves the second statement.

Note that the lemma proved shows in particular that any left differential module is a left module over the algebra Diff(A) which justifies our terminology.

Due to the above result, any complex of differential operators ⋯ ⟶ Q_{i} ⟶ Q_{i+1} ⟶ ⋯ and a left differential module P generate the complex ⋯ ⟶ Q_{i} ⊗_{A} P ⟶ Q_{i+1} ⊗_{A} P ⟶ ⋯ “with coefficients” in P. In particular, since the co-gluing c^{∞,∞} is in an obvious way co-associative, i.e., the diagram

J^{∞}(P)  ⟶^{c^{∞,∞}(P)}  J^{∞}(J^{∞}(P))
c^{∞,∞}(P)↓  ↓^{J^{∞}(c^{∞,∞}(P))}
J^{∞}(J^{∞}(P)) ⟶^{c^{∞,∞}(J^{∞}(P))} J^{∞}(J^{∞}(J^{∞}(P)))

is commutative, J^{∞}(P) is a left differential module with λ = c^{∞,∞}. Consequently, we can consider the de Rham complex with coefficients in J^{∞}(P):

0 ⟶ P ⟶^{j_{∞}} J^{∞}(P) ⟶ Λ^{1} ⊗_{A} J^{∞}(P) ⟶ ⋯

⋯ ⟶ Λ^{i} ⊗_{A} J^{∞}(P) ⟶ Λ^{i+1} ⊗_{A} J^{∞}(P) ⟶ ⋯

which is the inverse limit for the Jet-Spencer complexes of P

0 ⟶ P ⟶^{j_{k}} J^{k}(P) ⟶^{S} Λ^{1} ⊗_{A} J^{k−1}(P) ⟶^{S} ⋯

⋯ ⟶^{S} Λ^{i} ⊗_{A} J^{k−i}(P) ⟶^{S} Λ^{i+1} ⊗_{A} J^{k−i−1}(P) ⟶ ⋯,

where S(ω ⊗ j_{k−i}(p)) = dω ⊗ j_{k−i−1}(p).

Let Δ: P → Q be a differential operator and ψ_{∞}^{Δ} : J^{∞}(P) → J^{∞}(Q) be the corresponding homomorphism. The kernel E_{Δ} = ker ψ_{∞}^{Δ} inherits the left differential module structure of J^{∞}(P) and we can consider the de Rham complex with coefficients in E_{Δ}:

0 ⟶ E_{Δ} ⟶ Λ^{1} ⊗_{A} E_{Δ} ⟶ ⋯ ⟶ Λ^{i} ⊗_{A} E_{Δ} ⟶ Λ^{i+1} ⊗_{A} E_{Δ} ⟶ ⋯ (1.10)

which is called the Jet-Spencer complex of the operator Δ.

Now we shall introduce the concept dual to that of left differential modules.

Definition 1.8. An A-module P is called a right differential module, if there exists an A-module homomorphism ρ: Diff^{+}(P) → P that satisfies the equality ρ |_{Diff_{0}^{+}(P)} = id_{P} and makes the diagram

Diff^{+}(Diff^{+}(P)) ⟶^{c_{∞,∞}} Diff^{+}(P)
Diff^{+}(ρ)↓  ↓^{ρ}
Diff^{+}(P)  ⟶^{ρ}  P

commutative.

Lemma 1.17. Let P be a right differential module. Then for any differential operator Δ: Q_{1} → Q_{2} of order ≤ k there exists an operator

Δ^{P} : Hom_{A}(Q_{2}, P) → Hom_{A}(Q_{1}, P)

of order ≤ k satisfying id_{Q}^{P} = id_{Hom_{A}(Q,P)} for Q = Q_{1} = Q_{2} and

(Δ_{2} ∘ Δ_{1})^{P} = Δ_{1}^{P} ∘ Δ_{2}^{P}

for any operators Δ_{1} : Q_{1} → Q_{2}, Δ_{2} : Q_{2} → Q_{3}.

Proof. Let us define the action of Δ^{P} by setting Δ^{P} (f) = ρ ∘ ψ_{f∘Δ}, where f ∈ Hom_{A}(Q_{2}, P). Obviously, this is a k-th order differential operator and id_{Q}^{P} = id_{Hom_{A}(Q,P)}. Now,

(Δ_{2} ∘ Δ_{1})^{P} = ρ ∘ ψ_{f∘Δ_{2}∘Δ_{1}} = ρ ∘ c_{∞,∞} ∘ Diff^{+}(ψ_{f∘Δ_{2}}) ∘ ψ_{Δ_{1}}
= ρ ∘ Diff^{+}(ρ ∘ ψ_{f∘Δ_{2}}) ∘ ψ_{Δ_{1}} = ρ ∘ Diff^{+}(Δ_{2}^{P} (f)) ∘ ψ_{Δ_{1}}
= Δ_{1}^{P} (Δ_{2}^{P} (f)).

Hence, (·)^{P} preserves composition.

From the lemma proved it follows that any right differential module is a right module over the algebra Diff(A).

Let ⋯ → Q_{i} ⟶^{Δ_{i}} Q_{i+1} → ⋯ be a complex of differential operators and P be a right differential module. Then, by Lemma 1.17, we can construct

the dual complex ⋯ ⟵ Hom_{A}(Q_{i}, P) ⟵^{Δ_{i}^{P}} Hom_{A}(Q_{i+1}, P) ⟵ ⋯ with coefficients in P. Note that the Diff-Spencer complex is a particular case of this construction. In fact, due to properties of the homomorphism c_{∞,∞} the module Diff^{+}(P) is a right differential module with ρ = c_{∞,∞}. Applying Lemma 1.17 to the de Rham complex, we obtain the Diff-Spencer complex.

Note also that if Δ: P → Q is a differential operator, then the cokernel C_{Δ} of the homomorphism ψ_{Δ}^{∞}: Diff^{+}(P) → Diff^{+}(Q) inherits the right differential module structure of Diff^{+}(Q). Thus we can consider the complex

0 ⟵ coker Δ ⟵^{D} C_{Δ} ⟵ D_{1}(C_{Δ}) ⟵ ⋯ ⟵ D_{i}(C_{Δ}) ⟵ D_{i+1}(C_{Δ}) ⟵ ⋯

dual to the de Rham complex with coefficients in C_{Δ}. It is called the Diff-Spencer complex of the operator Δ.

1.7. The Spencer cohomology. Consider an important class of commutative algebras.

Definition 1.9. An algebra A is called smooth, if the module Λ^{1} is projective and of finite type.

In this section we shall work over a smooth algebra A.

Take two Diff-Spencer complexes, of orders k and k − 1, and consider their embedding

0 ⟵ P ⟵ Diff_{k}^{+}(P)) ⟵ D_{1}(Diff_{k−1}^{+}(P)) ⟵ ⋯
∥  ↑  ↑
0 ⟵ P ⟵ Diff_{k−1}^{+}(P)) ⟵ D_{1}(Diff_{k−2}^{+}(P)) ⟵ ⋯

Then, if the algebra A is smooth, the direct sum of the corresponding quotient complexes is of the form

0 ⟵ Smbl(A, P) ⟵^{δ} D_{1}(Smbl(A, P)) ⟵^{δ} D_{2}(Smbl(A, P)) ⟵ ⋯

By standard reasoning, exactness of this complex implies that of Diffcomplexes.

Exercise 1.5. Prove that the operators δ are A-homomorphisms.

Let us describe the structure of the modules Smbl(A, P). For the time being, use the notation D = D_{1}(A). Consider the homomorphism α_{k}: P ⊗_{A} S^{k}(D) → Smbl_{k}(A, P) defined by

α_{k}(p ⊗ ∇_{1} · · · · · ∇_{k}) = smbl_{k}(Δ), Δ(a) = (∇_{1} ∘ ⋯ ∘ ∇_{k})(a)p,

where a ∈ A, p ∈ P, and smbl_{k}: Diff_{k}(A, P) ⟶ Smbl_{k}(A, P) is the natural projection.

Lemma 1.18. If A is a smooth algebra, the homomorphism α_{k} is an isomorphism.

Proof. Consider a differential operator Δ: A → P of order ≤ k. Then the map s_{Δ} : A^{⊗k} → P defined by s_{Δ}(a_{1}, …, a_{k}) = δ_{a_{1},...,a_{k}}(Δ) is a symmetric multiderivation and thus the correspondence Δ ↦ s_{Δ} generates a homomorphism

Smbl_{k}(A, P) → Hom_{A}(S^{k}(Λ^{1}), P) = S^{k}(D) ⊗_{A} P,  (1.11)

which, as it can be checked by direct computation, is inverse to α_{k}. Note that the second equality in (1.11) is valid because A is a smooth algebra.

Exercise 1.6. Prove that the module Smbl_{k}(P, Q) is isomorphic to the module S^{k}(D) ⊗_{A} Hom_{A}(P, Q).

Exercise 1.7. Dualize Lemma 1.18. Namely, prove that the kernel of the natural projection ν_{k,k−1} : J^{k}(P) → J^{k−1}(P) is isomorphic to S^{k}(Λ^{1})⊗_{A} P, with the isomorphism α^{k}: S^{k}(Λ^{1}) ⊗_{A} P → ker ν_{k,k−1} given by

α^{k}(da_{1} · . . . · da_{k} ⊗ p) = δ_{a_{1},...,a_{k}}(j_{k})(p), p ∈ P.

Thus we obtain:

D_{i}(Smbl_{k}(P)) = Hom_{A}(Λ^{i}, P ⊗_{A} S^{k}(D)) = P ⊗_{A} S^{k}(D) ⊗_{A} Λ^{i}(D).

But from the definition of the Spencer operator it easily follows that the action of the operator

δ: P ⊗_{A} S^{k}(D) ⊗_{A} Λ^{i}(D) → P ⊗_{A} S^{k+1}(D) ⊗_{A} Λ^{i−1}(D)

is expressed by

δ(p ⊗ σ ⊗ ∇_{1} ∧ ⋯ ∧ ∇_{i})

= ∑_{l=1}^{i}(−1)^{l+1}p ⊗ σ · ∇_{l} ⊗ ∇_{1} ∧ ⋯ ∧ ˆ∇_{l} ∧ ⋯ ∧ ∇_{i}

where p ∈ P, σ ∈ S^{k}(D), ∇_{l} ∈ D and the “hat” means that the corresponding term is omitted. Thus we see that the operator δ coincides with the Koszul differential (see the Appendix) which implies exactness of Diff-Spencer complexes.

The Jet-Spencer complexes are dual to them and consequently, in the situation under consideration, are exact as well. This can also be proved independently by considering two Jet-Spencer complexes of orders k and k − 1 and their projection

0 ⟶ P ⟶ J^{k}(P)) ⟶ Λ^{1} ⊗_{A} J^{k−1}(P) ⟶ ⋯
∥  ↓  ↓
0 ⟶ P ⟶ J^{k−1}(P)) ⟶ Λ^{1} ⊗_{A} J^{k−2}(P) ⟶ ⋯

Then the corresponding kernel complexes are of the form

0 ⟶ S^{k}(Λ^{1}) ⊗_{A} P ⟶^{δ} Λ^{1} ⊗_{A} S^{k−1}(Λ^{1}) ⊗_{A} P

⟶^{δ} Λ^{2} ⊗_{A} S^{k−2}(Λ^{1}) ⊗_{A} P ⟶ ⋯

and are called the δ-Spencer complexes of P. These are complexes of Ahomomorphisms. The operator

δ: Λ^{s} ⊗_{A} S^{k−s}(Λ^{1}) ⊗_{A} P → Λ^{s+1} ⊗_{A} S^{k−s−1}(Λ^{1}) ⊗_{A} P

is defined by δ(ω ⊗ u ⊗ p) = (−1)^{s}ω ∧ i(u) ⊗ p, where i: S^{k−s}(Λ^{1}) → Λ^{1} ⊗ S^{k−s−1}(Λ^{1}) is the natural inclusion. Dropping the multiplier P we get the de Rham complexes with polynomial coefficients. This proves that the δ-Spencer complexes and, therefore, the Jet-Spencer complexes are exact.

Thus we have the following

Theorem 1.19. If A is a smooth algebra, then all Diff-Spencer complexes and Jet-Spencer complexes are exact.

Now, let us consider an operator Δ: P → P_{1} of order ≤ k. Our aim is to compute the Jet-Spencer cohomology of Δ, i.e., the cohomology of the complex (1.10) on page 17.

Definition 1.10. A complex of C-differential operators ⋯ ⟶ P_{i−1} ⟶^{Δ_{i}} P_{i} ⟶^{Δ_{i+1}} P_{i+1} ⟶ ⋯ is called formally exact, if the complex

⋯ ⟶ J̄^{k_{i}+k_{i+1}+l}(P_{i−1}) ⟶^{φ_{Δ_{i}}^{k_{i}+k_{i+1}+l}} J̄^{k_{i+1}+l}(P_{i}) ⟶^{φ_{Δ_{i+1}}^{k_{i+1}+l}} J̄^{l}(P_{i+1}) ⟶ ⋯,

with ord Δ_{j} ≤ k_{j}, is exact for any l.

Theorem 1.20. Jet-Spencer cohomology of Δ coincides with the cohomology of any formally exact complex of the form

0 ⟶ P ⟶^{Δ} P_{1} ⟶ P_{2} ⟶ P_{3} ⟶ ⋯

Proof. Consider the following commutative diagram

⋮  ⋮  ⋮
↑  ↑  ↑
0 ⟶ Λ^{2} ⊗ J^{∞}(P) ⟶ Λ^{2} ⊗ J^{∞}(P_{1}) ⟶ Λ^{2} ⊗ J^{∞}(P_{2}) ⟶ ⋯
↑d̄  ↑d̄  ↑d̄
0 ⟶ Λ^{1} ⊗ J^{∞}(P) ⟶ Λ^{1} ⊗ J^{∞}(P_{1}) ⟶ Λ^{1} ⊗ J^{∞}(P_{2}) ⟶ ⋯
↑d̄  ↑d̄  ↑d̄
0 ⟶ J^{∞}(P) ⟶ J^{∞}(P_{1}) ⟶ J^{∞}(P_{2}) ⟶ ⋯
↑  ↑  ↑
0  0  0

where the i-th column is the de Rham complex with coefficients in the left differential module J^{∞}(P_{i}). The horizontal maps are induced by the operators Δ_{i}. All the sequences are exact except for the terms in the left column and the bottom row. Now the standard spectral sequence arguments (see the Appendix) completes the proof.

Our aim now is to prove that in a sense all compatibility complexes are formally exact. To this end, let us discuss the notion of involutiveness of a differential operator.

The map ψ_{l}^{Δ} : J^{k+l}(P) → J^{l}(P_{1}) gives rise to the map

smbl_{k,l}(Δ): S^{k+l}(Λ^{1}) ⊗ P → S^{l}(Λ^{1}) ⊗ P_{1}

called the l-th prolongation of the symbol of Δ.

Exercise 1.8. Check that 0-th prolongation map smbl_{k,0} : Diff_{k}(P, P_{1}) → Hom(S^{k}(Λ^{1}) ⊗ P, P_{1}) coincides with the natural projection of differential operators to their symbols, smbl_{k}: Diff_{k}(P, P_{1}) → Smbl_{k}(P, P_{1}).

Consider the symbolic module g^{k+l} = ker smbl_{k,l}(Δ) ⊂ S^{k+l}(Λ^{1}) ⊗ P of the operator Δ. It is easily shown that the subcomplex of the δ-Spencer complex

0 ⟶ g^{k+l} ⟶^{δ} Λ^{1} ⊗ g^{k+l−1} ⟶^{δ} Λ^{2} ⊗ g^{k+l−2} ⟶^{δ} ⋯  (1.12)

is well defined. The cohomology of this complex in the term Λ^{i} ⊗ g^{k+l−i} is denoted by H^{k+l,i}(Δ) and is said to be δ-Spencer cohomology of the operator Δ.

Exercise 1.9. Prove that H^{k+l,0}(Δ) = H^{k+l,1}(Δ) = 0.

The operator Δ is called involutive (in the sense of Cartan), if H^{k+l,i}(Δ) = 0 for all i ≥ 0.

Definition 1.11. An operator Δ is called formally integrable, if for all l modules E_{Δ}^{l} = ker ψ_{Δ}^{l} ⊂ J^{k+l}(P) and g^{k+l} are projective and the natural mappings E_{Δ}^{l} → E_{Δ}^{l−1} are surjections.

Till the end of this section we shall assume all the operators under consideration to be formally integrable.

Theorem 1.21. If the operator Δ is involutive, then the compatibility complex of Δ is formally exact for all positive integers k_{1}, k_{2}, k_{3}, ….

Proof. Suppose that the compatibility complex of Δ

P ⟶^{Δ} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} ⋯

is formally exact in terms P_{1}, P_{2}, …, P_{i−1}. The commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶ g^{K} ⟶ S^{K} ⊗ P ⟶ S^{K−k} ⊗ P_{1} ⟶ ⋯
↓  ↓  ↓
0 ⟶ E_{Δ}^{K−k} ⟶ J^{K}(P) ⟶ J^{K−k}(P_{1}) ⟶ ⋯
↓  ↓  ↓
0 ⟶ E_{Δ}^{K−k−1} ⟶ J^{K−1}(P) ⟶ J^{K−k−1}(P_{1}) ⟶ ⋯
↓  ↓  ↓
0  0  0
0  0
↓  ↓
⋯ ⟶ S^{k_{i}} ⊗ P_{i} ⟶ P_{i+1} ⟶ 0
↓  ↓
⋯ ⟶ J^{k_{i}}(P_{i}) ⟶ P_{i+1} ⟶ 0
↓  ↓
⋯ ⟶ J^{k_{i}−1}(P_{i}) ⟶ 0

↓

0

where S^{j} = S^{j}(Λ^{1}), K = k + k_{1} + k_{2} + ⋯ + k_{i}, shows that the complex

0 ⟶ g^{K} ⟶ S^{K} ⊗ P ⟶ S^{K−k} ⊗ P_{1} ⟶ ⋯ ⟶ S^{k_{i}} ⊗ P_{i}

is exact.

What we must to prove is that the sequences

S^{k_{i−1}+k_{i}+l} ⊗ P_{i−1} ⟶ S^{k_{i}+l} ⊗ P_{i} ⟶ S^{l} ⊗ P_{i+1}

are exact for all l ≥ 1. The proof is by induction on l, with the inductive step involving the standard spectral sequence arguments applied to the commutative diagram

0 ⟶ S^{l} ⊗ P_{i+1} ⟶^{δ} Λ^{1} ⊗ S^{l−1} ⊗ P_{i+1} ⟶^{δ} Λ^{2} ⊗ S^{l−2} ⊗ P_{i+1} ⟶^{δ} ⋯
↑  ↑  ↑
0 ⟶ S^{k_{i}+l} ⊗ P_{i} ⟶^{δ} Λ^{1} ⊗ S^{k_{i}+l−1} ⊗ P_{i} ⟶^{δ} Λ^{2} ⊗ S^{k_{i}+l−2} ⊗ P_{i} ⟶^{δ} ⋯
↑  ↑  ↑
⋮  ⋮  ⋮
↑  ↑  ↑
0 ⟶ S^{K+l} ⊗ P_{0} ⟶^{δ} Λ^{1} ⊗ S^{K+l−1} ⊗ P_{0} ⟶^{δ} Λ^{2} ⊗ S^{K+l−2} ⊗ P_{0} ⟶^{δ} ⋯
↑  ↑  ↑
0 ⟶ g^{K+l}  ⟶^{δ} Λ^{1} ⊗ g^{K+l−1}  ⟶^{δ} Λ^{2} ⊗ g^{K+l−2}  ⟶^{δ} ⋯
↑  ↑  ↑
0  0  0

Example 1.1. For the de Rham differential d: A → Λ^{1} the symbolic modules g^{l} are trivial. Hence, the de Rham differential is involutive and, therefore, the de Rham complex is formally exact.

Example 1.2. Consider the geometric situation and suppose that the manifold M is a (pseudo-)Riemannian manifold. For an integer p consider the operator Δ = d∗d: Λ^{p} → Λ^{n−p}, where ∗ is the Hodge star operator on the modules of differential forms. Let us show that the complex

Λ̄^{p} ⟶^{Δ} Λ̄^{n−p} ⟶^{d} Λ̄^{n−p+1} ⟶^{d} Λ^{n−p+2} ⟶^{d} ⋯ ⟶^{d} Λ^{n} ⟶ 0

is formally exact and, thus, is the compatibility complex for the operator Δ. In view of the previous example we must prove that the image of the map smbl(Δ): S^{l+2} ⊗ Λ^{p} → S^{l} ⊗ Λ^{n−p} coincides with the image of the map smbl(d): S^{l+1} ⊗ Λ^{n−p−1} → S^{l} ⊗ Λ^{n−p} for all l ≥ 0. Since Δ∗ = d∗d∗ = d(∗d∗ + d), it is sufficient to show that the map smbl(∗d∗ + d): S^{l+1} ⊗ (Λ^{n−p+1} ⊕ Λ^{n−p−1}) → S^{l} ⊗ Λ^{n−p} is an epimorphism. Consider smbl(L): S^{l} ⊗ Λ^{n−p} → S^{l} ⊗ Λ^{n−p}, where L = (∗d∗ + d)(∗d∗ ± d) is the Laplace operator. From coordinate considerations it easily follows that the symbol of the Laplace operator is epimorphic, and so the symbol of the operator ∗d∗ + d is also epimorphic.

The condition of involutiveness is not necessary for the formal exactness of the compatibility complex due to the following Theorem 1.22 (δ-Poincaré lemma). If the algebra A is Noetherian, then for any operator Δ there exists an integer l_{0} = l_{0}(m, n, k), where m = rank P, such that H^{k+l,i}(Δ) = 0 for l ≥ l_{0} and i ≥ 0.

Proof can be found, e.g., in [32, 10]. Thus, from the proof of Theorem 1.21 on page 22 we see that for sufficiently large integer k_{1} the compatibility complex is formally exact for any operator Δ.

We shall always assume that compatibility complexes are formally exact.

1.8. Geometrical modules. There are several directions to generalize or specialize the above described theory. Probably, the most important one, giving rise to various interesting specializations, is associated with the following concept.

Definition 1.12. An abelian subcategory M(A) of the category of all Amodules is said to be differentially closed, if

(1) it is closed under tensor product over A,

(2) it is closed under the action of the functors Diff_{k}^{(+)}(·, ·) and D_{i}(·),

(3) the functors Diff_{k}^{(+)}(P, ·), Diff_{k}^{(+)}(·, Q) and D_{i}(·) are representable in M(A), whenever P, Q are objects of M(A).

As an example consider the following situation. Let M be a smooth (i.e., C^{∞}-class) finite-dimensional manifold and set A = C^{∞}(M). Let π : E → M, ξ : F → M be two smooth locally trivial finite-dimensional vector bundles over M and P = Γ(π), Q = Γ(ξ) be the corresponding A-modules of smooth sections.

One can prove that the module Diff_{k}^{(+)}(P, Q) coincides with the module of k-th order differential operators acting from the bundle π to ξ (see Proposition 1.1 on page 6). Further, the module D(A) coincides with the module of vector fields on the manifold M.

However if one constructs representative objects for the functors such as Diff_{k}(P, ·) and D_{i}(·) in the category of all A-modules, the modules J^{k}(P) and Λ^{i} will not coincide with “geometrical” jets and differential forms.

Exercise 1.10. Show that in the case M = ℝ the form d(sin x) − cos x dx is nonzero.

Definition 1.13. A module P over C^{∞}(M) is called geometrical, if

⋂_{x∈M} μ_{x}P = 0,

where μ_{x} is the ideal in C^{∞}(M) consisting of functions vanishing at point x ∈ M.

Denote by G(M) the full subcategory of the category of all modules whose objects are geometrical C^{∞}(M)-modules. Let P be an A-module and set

G(P) = P/⋂_{x∈M} μ_{x}P.

Evidently, G(P) is a geometrical module while the correspondence P ⇒ G(P) is a functor from the category of all C^{∞}(M)-modules to the category G(M) of geometrical modules.

Proposition 1.23. Let M be a smooth finite-dimensional manifold and A = C^{∞}(M). Then

(1) The category G(A) of geometrical A-modules is differentially closed.

(2) The representative objects for the functors Diff_{k}(P, ·) and D_{i}(·) in G(A) coincide with G(J^{k}(P)) and G(Λ^{i}) respectively.

(3) The module G(Λ^{i}) coincides with the module of differential i-forms on M.

(4) If P = Γ(π) for a smooth locally trivial finite-dimensional vector bundle π : E → M, then the module G(J^{k}(P)) coincides with the module Γ(π_{k}), where π_{k} : J^{k}(π) → M is the bundle of k-jets for the bundle π (see Section 3.1).

Exercise 1.11. Prove (1), (2), and (3) above.

The situation described in this Proposition will be referred to as the geometrical one.

Another example of a differentially closed category is the category of filtered geometrical modules over a filtered algebra. This category is essential to construct differential calculus over manifolds of infinite jets and infinitely prolonged differential equations (see Sections 3.3 and 3.8 respectively).

Remark 1.5. The logical structure of the above described theory is obviously generalized to the supercommutative case. For a noncommutative generalization see [54, 55].

2. Algebraic model for Lagrangian formalism

Using the above introduced algebraic concepts, we shall construct now an algebraic model for Lagrangian formalism; see also [53]. For geometric motivations, we refer the reader to Section 7 and to Subsection 7.5 especially.

2.1. Adjoint operators. Consider an A-module P and the complex of A-homomorphisms

0 ⟶ Diff^{+}(P, A) ⟶^{w} Diff^{+}(P, Λ^{1}) ⟶^{w} Diff^{+}(P, Λ^{2}) ⟶^{w} ⋯,  (2.1)

where, by definition, w(∇) = d ∘ ∇ ∈ Diff^{+}(P, Λ^{i+1}) for the operator ∇ ∈ Diff^{+}(P, Λ^{i}). Let P̂_{n}, n ≥ 0, be the cohomology module of this complex at the term Diff^{+}(P, Λ^{n}).

Any operator Δ: P → Q determines the natural cochain map

⋯ ⟶ Diff^{+}(Q, Λ^{i−1}) ⟶^{w} Diff^{+}(Q, Λ^{i}) ⟶ ⋯
Δ̃↓  Δ̃↓
⋯ ⟶ Diff^{+}(P, Λ^{i−1}) ⟶^{w} Diff^{+}(P, Λ^{i}) ⟶ ⋯

where Δ̃(∇) = ∇ ∘ Δ ∈ Diff^{+}(P, Λ^{i}) for ∇ ∈ Diff^{+}(Q, Λ^{i}).

Definition 2.1. The cohomology map Δ_{n}^{∗}: Q̂_{n} → P̂_{n} induced by Δ̃ is called the (n-th) adjoint operator for Δ.

Below we assume n to be fixed and omit the corresponding subscript. The main properties of the adjoint operator are described by

Proposition 2.1. Let P, Q and R be A-modules. Then (1) If Δ ∈ Diff_{k}(P, Q), then Δ^{∗} ∈ Diff_{k}(Q̂, P̂).

(2) If Δ_{1} ∈ Diff(P, Q) and Δ_{2} ∈ Diff(Q, R), then (Δ_{2} ∘ Δ_{1})^{∗} = Δ_{1}^{∗} ∘ Δ_{2}^{∗}.

Proof. Let [∇] denote the cohomology class of ∇ ∈ Diff^{+}(P, Λ^{n}), where w(∇) = 0.

(1) Let a ∈ A. Then

δ_{a}(Δ^{∗})([∇]) = Δ^{∗}([∇]) − Δ^{∗}(a[∇]) = [∇ ∘ a ∘ Δ] − [∇ ∘ Δ ∘ a]
= (a ∘ Δ)^{∗}([∇]) − (Δ ∘ a)^{∗}([∇]) = −δ_{a}(Δ^{∗})([∇]).

Consequently, δ_{a_{0},...,a_{k}}(Δ^{∗}) = (−1)^{k+1}(δ_{a_{0},...,a_{k}}(Δ))^{∗} for any a_{0}, …, a_{k} ∈ A. (2) The second statement is implied by the following identities:

(Δ_{2} ∘ Δ_{1})^{∗}([∇]) = [∇ ∘ Δ_{2} ∘ Δ_{1}] = Δ_{1}^{∗}([∇ ∘ Δ_{2}]) = Δ_{1}^{∗}(Δ_{2}^{∗}([∇])),

which concludes the proof.

Example 2.1. Let a ∈ A and a = a_{P} : P → P be the operator of multiplication by a: p ↦ ap. Then obviously a_{P}^{∗} = aP̂.

Example 2.2. Let p ∈ P and p: A → P be the operator acting by a ↦ ap. Then, by Proposition 2.1 (1) on the preceding page, p^{∗} ∈ Hom_{A}(P̂, Â). Thus there exists a natural paring ⟨·, ·⟩: P ⊗_{A} P̂ → Â defined by ⟨p, p̂⟩ = p^{∗}(p̂), p̂ ∈ P̂.

2.2. Berezinian and integration. Consider a complex of differential operators ⋯ ⟶ P_{k} ⟶^{Δ_{k}} P_{k+1} ⟶ · · ·. Then, by Proposition 2.1 on the page

before, ⋯ ⟵ P̂_{k} ⟵^{Δ_{k}^{∗}} P̂_{k+1} ⟵ ⋯ is a complex of differential operators as well. This complex called adjoint to the initial one.

Definition 2.2. The complex adjoint to the de Rham complex of the algebra A is called the complex of integral forms and is denoted by

0 ⟵ Σ_{0} ⟵^{δ} Σ_{1} ⟵^{δ} ⋯,

where Σ_{i} = Λ̂^{i}, δ = d^{∗}. The module Σ_{0} = Â is called the Berezinian (or the module of the volume forms) and is denoted by B.

Assume that the modules under consideration are projective and of finite type. Then we have P̂ = Hom_{A}(P, B). In particular, Σ_{i} = Λ̂^{i} = D_{i}(B).

Let us calculate the Berezinian in the geometrical situation (see Subsection 1.8), when A = C^{∞}(M).

Theorem 2.2. If A = C^{∞}(M), M being a smooth finite-dimensional manifold, then

(1) Â_{s} = 0 for s ≠ n = dim M.

(2) Â_{n} = B = Λ^{n}, i.e., the Berezinian coincides with the module of forms of maximal degree. This isomorphism takes each form ω ∈ Λ^{n} to the cohomology class of the zero-order operator ω: A → Λ^{n}, f ↦ fω.

The proof is similar to that of Theorem 1.19 on page 20 and is left to the reader.

In the geometrical situation there exists a natural isomorphism Λ^{i} → D_{n−i}(Λ^{n}) = Σ_{i} which takes ω ∈ Λ^{i} to the homomorphism ω: Λ^{n−i} → Λ^{n} defined by ω(η) = η ∧ ω, η ∈ Λ^{n−i}.

Exercise 2.1. Show that ⟨ω_{1}, ω_{2}⟩ = ω_{1} ∧ ω_{2}, ω_{1} ∈ Λ^{i}, ω_{2} ∈ Λ^{n−i}.

Exercise 2.2. Prove that d_{i}^{∗} = (−1)^{i+1}d_{n−i−1}, where d_{i}: Λ^{i} → Λ^{i+1} is the de Rham differential.

Thus, in the geometrical situation the complex of integral forms coincides (up to a sign) with the de Rham complex.

Exercise 2.3. Prove the coordinate formula for the adjoint operator:

(1) if Δ = ∑_{σ} a_{σ} \frac{∂^{|σ|}}{∂x_{σ}} is a scalar operator, then Δ^{∗} = ∑_{σ}(−1)^{|σ|} \frac{∂^{|σ|}}{∂x_{σ}} ∘ a_{σ}; (2) if Δ = ∥Δ_{ij}∥ is a matrix operator, then Δ^{∗} = ∥Δ_{ji}^{∗}∥.

The operator D: Diff^{+}(Λ^{k}) → Λ^{k} defined on page 8 generates the map ∫ : B → H^{∗}(Λ^{•}) from the Berezinian to the de Rham cohomology group of A. Namely, for any operator ∇ ∈ Diff(A, Λ^{n}) satisfying d ∘ ∇ = 0 we set ∫ [∇] = [∇(1)], where [·] denotes the cohomology class.

Proposition 2.3. The map ∫ : B → H^{∗}(Λ^{•}) possesses the following properties:

(1) If ω ∈ Σ_{1}, then ∫ δω = 0.

(2) For any differential operator Δ: P → Q and elements p ∈ P, q̂ ∈ Q̂ the identity

∫ ⟨Δ(p), q̂⟩ = ∫ ⟨p, Δ^{∗}(q̂)⟩

holds.

Proof. (1) Let ω = [∇] ∈ Σ_{1}. Then δω = [∇ ∘ d] and consequently ∫ ω = [∇d(1)] = 0.

(2) Let q̂ = [∇] for some operator ∇: Q → Λ^{n}. Then

∫ ⟨Δ(p), q̂⟩ = ∫ [∇Δ(p)] = ∫ ∇ ∘ Δ ∘ p

= ∫ ⟨p, [∇ ∘ Δ]⟩ = ∫ ⟨p, Δ^{∗}(q̂)⟩,

which completes the proof.

Remark 2.1. Note that the Berezinian B is a differential right module (see Subsection 1.6) and the complex of integral forms may be understood as the complex dual to the de Rham complex with coefficients in B.

Exercise 2.4. Show that in the geometrical situation the right action of vector fields can also be defined via X(ω) = −L_{X}(ω), where L_{X} is the Lie derivative.

Now we establish a relationship between the de Rham cohomology and the homology of the complex of integral forms.

Proposition 2.4 (algebraic Poincaré duality). There exists a spectral sequence (E_{p,q}^{r}, d_{p,q}^{r}) with

E_{p,q}^{2} = H_{p}((Σ_{•})_{−q}),

the homology of complexes of integral forms, and converging to the de Rham cohomology H(Λ^{•}).

Proof. Consider the commutative diagram

0  0  0
↑  ↑  ↑
0 ⟶ Diff^{+}(A, A) ⟶^{w} Diff^{+}(A, Λ^{1}) ⟶^{w} Diff^{+}(A, Λ^{2}) ⟶ ⋯
↑d̃  ↑d̃  ↑d̃
0 ⟶ Diff^{+}(Λ^{1}, A) ⟶^{w} Diff^{+}(Λ^{1}, Λ^{1}) ⟶^{w} Diff^{+}(Λ^{1}, Λ^{2}) ⟶ ⋯
↑d̃  ↑d̃  ↑d̃
0 ⟶ Diff^{+}(Λ^{2}, A) ⟶^{w} Diff^{+}(Λ^{2}, Λ^{1}) ⟶^{w} Diff^{+}(Λ^{2}, Λ^{2}) ⟶ ⋯
↑d̃  ↑d̃  ↑d̃
⋮  ⋮  ⋮

where the differential d̃: Diff^{+}(Λ^{k+1}, P) → Diff^{+}(Λ^{k}, P) is defined by d̃(Δ) = Δ ∘ d. The statement follows easily from the standard spectral sequence arguments.

2.3. Green’s formula. Let Q be an A-module. Then a natural homomorphism ξ_{Q}: Q → ˆ Q̂ defined by ξ_{Q}(q)(q̂) = ⟨q, q̂⟩ exists. Consequently, to any operator Δ: P → Q̂ there corresponds the operator Δ^{∘}: Q → P̂, where Δ^{∘} = Δ^{∗} ∘ ξ_{Q}. This operator will also be called adjoint to Δ.

Remark 2.2. In the geometrical situation the two notions of adjointness coincide.

Example 2.3. Let q̂ ∈ Q̂ and q̂: A → Q̂ be the zero-order operator defined by a ↦ aq̂. The adjoint operator is q̂ itself understood as an element of Hom_{A}(Q, B).

Proposition 2.5. The correspondence Δ ↦ Δ^{∘} possesses the following properties:

(1) Let Δ ∈ Diff(P, Q̂) and Δ(p) = [∇_{p}], where ∇_{p} ∈ Diff(Q, Λ^{i}). Then Δ^{∘}(q) = [□_{q}], where □_{q} ∈ Diff(P, Λ^{i}) and □_{q}(p) = ∇_{p}(q).

(2) For any Δ ∈ Diff(P, Q̂), one has (Δ^{∘})^{∘} = Δ.

(3) For any a ∈ A, one has (aΔ)^{∘} = Δ^{∘} ∘ a.

(4) If Δ ∈ Diff_{k}(P, B), then Δ^{∘} = j_{k}^{∗} ∘ (aΔ).

(5) If X ∈ D_{1}(B), then X + X^{∘} = δX ∈ Diff_{0}(A, B) = B.

Proof. Statements (1), (3), and (4) are the direct consequences of the definition. Statement (2) is implied by (1). Let us prove (5).

Evidently, δ_{a}(j_{1}) = j_{1}(a) − aj_{1}(1) ∈ J^{1}(A). Hence for an operator Δ ∈ Diff_{1}(A, P) one has (δ_{a}(j_{1}))^{∗}(Δ) = Δ(a)−aΔ(1) = (δ_{a}Δ)(1). Consequently,

δ_{a}(X + X^{∘})(1) = (δ_{a}X)(1) + (δ_{a}(j_{1}^{∗}))(X) = (δ_{a}X)(1) − δ_{a}(j_{1})^{∗}(X) = 0

and finally δX = j_{1}^{∗}(X) = X^{∘}(1) = X + X^{∘}.

Note that Statements (1) and (4) of Proposition 2.5 on the facing page can be taken for the definition of Δ^{∘}.

Note now that from Proposition 1.15 on page 15 it follows that the modules D_{i}(P), i ≥ 2, can be described as

D_{i}(P) = { ∇ ∈ Diff_{1}(Λ^{i−1}, P) | ∇ ∘ d = 0 }.

Taking B for P, one can easily show that δ∇ = ∇^{∘}(1) and the last equality holds for i = 1 as well. Proposition 2.5 on the facing page shows that the correspondence Δ ↦ Δ^{∘} establishes an isomorphism between the modules Diff(P, Q̂) and Diff^{+}(Q, P̂) which, taking into account Proposition 1.15 on page 15, means that the Diff-Spencer complex of the module P̂ is isomorphic to the complex

0 ⟵ P̂ ⟵^{μ} Diff(P, B) ⟵^{ω} Diff(P, Σ_{1}) ⟵^{ω} Diff(P, Σ_{2}) ⟵ ⋯,  (2.2)

where ω(∇) = δ ∘ ∇, μ(∇) = ∇^{∘}(1). From Theorem 1.19 on page 20 one immediately obtains

Theorem 2.6. Complex (2.2) is exact.

Remark 2.3. Let Δ: P → Q be a differential operator. Then obviously the following commutative diagram takes place:

0 ⟵ Q̂ ⟵^{μ} Diff(Q, B) ⟵^{ω} Diff(Q, Σ_{1}) ⟵^{ω} ⋯
Δ^{∗}↓  ↓  ↓
0 ⟵ P̂ ⟵^{μ} Diff(P, B) ⟵^{ω} Diff(P, Σ_{1}) ⟵^{ω} ⋯

As a corollary of Theorem 2.6 we obtain

Theorem 2.7 (Green’s formula). If Δ ∈ Diff(P, Q̂), p ∈ P, q ∈ Q, then

⟨q, Δ(p)⟩ − ⟨Δ^{∘}(q), p⟩ = δG

for some integral 1-form G ∈ Σ_{1}.

Proof. Consider an operator ∇ ∈ Diff(A, B). Then ∇ − ∇^{∘}(1) lies in ker μ and consequently there exists an operator □ ∈ Diff(A, Σ_{1}) satisfying ∇ − ∇^{∘}(1) = ω(□) = δ ∘ □. Hence, ∇(1) − ∇^{∘}(1) = δG, where G = □(1). Setting ∇(a) = ⟨q, Δ(ap)⟩ we obtain the result.

Remark 2.4. The integral 1-form G is dependent on p and q. Let us show that we can choose G in such a way that the map p × q ↦ G(p, q) is a bidifferential operator. Note first that the map ω: Diff^{+}(A, Σ_{1}) → Diff^{+}(A, B) is an A-homomorphism. Since the module Diff^{+}(A, B) is projective, there exists an A-homomorphism κ: im ω → Diff^{+}(A, Σ_{1}) such that ω ∘ κ = id. We can put □ = κ(∇ − ∇(1)). Thus G = κ(∇ − ∇(1))(1). This proves the required statement.

Remark 2.5. From algebraic point of view, we see that in the geometrical situations there is the multitude of misleading isomorphisms, e.q., B = Λ^{n}, Δ^{∘} = Δ^{∗}, etc. In generalized settings, for example, in supercommutative situation (see Subsection 7.9 on page 130), these isomorphisms disappear.

2.4. The Euler operator. Let P and Q be A-modules. Introduce the notation

Diff_{(k)}(P, Q) = \underbrace{Diff(P, …, Diff(P}_{k times}, Q) …)

and set Diff_{(∗)}(P, Q) = ⨁_{k=0}^{∞} Diff_{(k)}(P, Q). A differential operator ∇ ∈ Diff_{(k)}(P, Q) satisfying the condition

∇(p_{1}, …, p_{i}, p_{i+1}, …, p_{k}) = σ∇(p_{1}, …, p_{i+1}, p_{i}, …, p_{k})

is called symmetric, if σ = 1, and skew-symmetric, if σ = −1 for all i. The modules of symmetric and skew-symmetric operators will be denoted by Diff_{(k)}^{sym}(P, Q) and Diff_{(k)}^{alt}(P, Q), respectively. From Theorem 2.6 on the preceding page and Corollary 1.3 on page 7 it follows that for any k the complex

0 ⟵ Diff_{(k)}(P, B) ⟵^{ω} Diff_{(k)}(P, Σ_{1}) ⟵^{ω} Diff_{(k)}(P, Σ_{2}) ⟵^{ω} ⋯,  (2.3)

where ω(∇) = δ ∘ ∇, is exact in all positive degrees, while its 0-homology is of the form H_{0}(Diff_{(k)}(P, Σ_{•})) = Diff_{(k−1)}(P, P̂). This result can be refined in the following way.

Theorem 2.8. The symmetric

0 ⟵ Diff_{(k)}^{sym}(P, B) ⟵^{ω} Diff_{(k)}^{sym}(P, Σ_{1}) ⟵^{ω} Diff_{(k)}^{sym}(P, Σ_{2}) ⟵^{ω} ⋯ (2.4)

and skew-symmetric

0 ⟵ Diff_{(k)}^{alt}(P, B) ⟵^{ω} Diff_{(k)}^{alt}(P, Σ_{1}) ⟵^{ω} Diff_{(k)}^{alt}(P, Σ_{2}) ⟵^{ω} ⋯  (2.5)

are acyclic complexes in all positive degrees, while the 0-homologies denoted by L_{k}^{sym}(P) and L_{k}^{alt}(P) respectively are of the form

L_{k}^{sym} = { ∇ ∈ Diff_{(k−1)}^{sym}(P, P̂) | (∇(p_{1}, …, p_{k−2}))^{∘} = ∇(p_{1}, …, p_{k−2}) },
L_{k}^{alt} = { ∇ ∈ Diff_{(k−1)}^{alt}(P, P̂) | (∇(p_{1}, …, p_{k−2}))^{∘} = −∇(p_{1}, …, p_{k−2}) }

for k > 1 and

L_{1}^{sym}(P) = L_{1}^{alt}(P) = P̂.

Proof. We shall consider the case of symmetric operators only, since the case of skew-symmetric ones is proved in the same way exactly.

Obviously, the complex (2.4) is a direct summand in (2.3) on the facing page and due to this fact the only thing we need to prove is that the diagram

Diff_{(k−1)}(P, P̂) ⟵^{μ_{(k−1)}} Diff_{(k)}(P, B)
ρ′↓  ↓^{ρ}
Diff_{(k−1)}(P, P̂) ⟵^{μ_{(k−1)}} Diff_{(k)}(P, B)

is commutative. Here

μ_{(k−1)}(∇)(p_{1}, …, p_{k−1}) = (∇(p_{1}, …, p_{k−1}))^{∘}(1),
ρ(∇)(p_{1}, …, p_{k−1}, p_{k}) = ∇(p_{1}, …, p_{k}, p_{k−1}),
ρ′(∇)(p_{1}, …, p_{k−2}) = (∇(p_{1}, …, p_{k−2}))^{∘}.

Note that μ_{(k−1)} = Diff_{(k−1)}(μ), where μ is defined in (2.2) on page 31. To prove commutativity, it suffices to consider the case k = 2. Let ∇ ∈ Diff_{(2)}(P, B) and ∇(p_{1}, p_{2}) = [Δ_{p_{1},p_{2}}]. Then μ_{(1)}(∇)(p_{1}) = [Δ′_{p_{1}}], where Δ′_{p_{1}}(p_{2}) = Δ_{p_{1},p_{2}}(1). Further, ρ′(μ_{(1)}(∇)) = [Δ′′_{p_{1}}], where

Δ′′_{p_{1}}(p_{2}) = Δ′_{p_{2}}(p_{1}) = Δ_{p_{2},p_{1}}(1).

On the other hand, one has ρ(∇)(p_{1}, p_{2}) = ∇(p_{2}, p_{1}) and μ_{(1)}(ρ(∇))(p_{1}) = [□_{p_{1}}], where □_{p_{1}}(p_{2}) = Δ_{p_{2},p_{1}}(1).

Definition 2.3. The elements of the space Lag(P) = ⨁_{k=1}^{∞} L_{k}^{sym}(P) are called Lagrangians of the module P. An operator L ∈ Diff_{(∗)}^{sym}(P, B) is called a density of a Lagrangian L, if L = L mod im ω. The natural correspondence E: Diff_{(∗)}^{sym}(P, B) → Diff_{(∗)}^{sym}(P, P̂), L ↦ L is called the Euler operator, while operators of the form Δ = E(L) are said to be Euler–Lagrange operators.

Theorem 2.8 on the facing page implies the following

Corollary 2.9. For any projective A-module P one has:

(1) An operator Δ ∈ Diff_{(∗)}^{sym}(P, P̂) is an Euler–Lagrange operator if and only if Δ is self-adjoint, i.e., if Δ ∈ L_{∗}^{sym}(P).

(2) A density L ∈ Diff_{(∗)}^{sym}(P, B) corresponds to a trivial Lagrangian, i.e., E(L) = 0, if and only if L is a total divergence, i.e., L ∈ im ω.

2.5. Conservation laws. Denote by F the commutative algebra of nonlinear operators^{5} Diff_{(∗)}^{sym}(P, A). Then for any A-module Q one has

Diff_{(∗)}^{sym}(P, Q) = F ⊗_{A} Q.

Let Δ ∈ F ⊗_{A} Q be a differential operator and let us set F_{Δ} = F/𝔞, where 𝔞 denotes the ideal in F generated by the operators of the form □ ∘ Δ, □ ∈ Diff(Q, A).

Thus, fixing P, we obtain the functor Q ⇒ F ⊗_{A}Q and fixing an operator Δ ∈ Diff_{(∗)}(P, Q) we get the functor Q ⇒ F_{Δ}⊗_{A}Q acting from the category M_{A} to M_{F} and to M_{F_{Δ}} respectively, where M denotes the category of all modules over the corresponding algebra. These functors in an obvious way generate natural transformations of the functors Diff_{k}^{(+)}(·), D_{k}(·), etc., and of their representative objects J^{k}(P), Λ^{k}, etc. For example, to any operator ∇: Q_{1} → Q_{2} there correspond operators F ⊗ ∇: F ⊗_{A} Q_{1} → F ⊗_{A} Q_{2} and F_{Δ} ⊗ ∇: F_{Δ} ⊗_{A} Q_{1} → F_{Δ} ⊗_{A} Q_{2}.

These natural transformations allow us to lift the theory of linear differential operators from A to F and to restrict the lifted theory to F_{Δ}. They are in parallel to the theory of C-differential operators (see the next section).

The natural embeddings

Diff_{(k)}^{sym}(P, R) ↪ Diff_{(k−1)}^{sym}(P, Diff(P, R))

generate the map ℓ: F ⊗_{A} R → F ⊗_{A} Diff(P, R), φ ↦ ℓ_{φ}, which is called the universal linearization. Using this map, we can rewrite Corollary 2.9 (1) on page 33 in the form ℓ_{Δ} = ℓ_{Δ}^{∘} while the Euler operator is written as E(L) = ℓ_{L}^{∘}(1). Note also that ℓ_{φψ} = φℓ_{ψ} + ψℓ_{φ} for any φ, ψ ∈ F ⊗_{A} R.

Definition 2.4. The group of conservation laws for the algebra F_{Δ} (or for the operator Δ) is the first homology group of the complex of integral forms

0 ⟵ F_{Δ} ⊗_{A} B ⟵ F_{Δ} ⊗_{A} Σ_{1} ⟵ F_{Δ} ⊗_{A} Σ_{2} ⟵ ⋯  (2.6)

with coefficients in F_{Δ}.

5In geometrical situation, this algebra is identified with the algebra of polynomial functions on infinite jets (see the next section).

3. Jets and nonlinear differential equations. Symmetries

We expose here main facts concerning geometrical approach to jets (finite and infinite) and to nonlinear differential operators. We shall confine ourselves with the case of vector bundles, though all constructions below can be carried out—with natural modifications—for an arbitrary locally trivial bundle π (and even in more general settings). For further reading, the books [32, 34] together with the paper [62] are recommended.

3.1. Finite jets. Let M be an n-dimensional smooth, i.e., of the class C^{∞}, manifold and π: E → M be a smooth m-dimensional vector bundle over M. Denote by Γ(π) the C^{∞}(M)-module of sections of the bundle π. For any point x ∈ M we shall also consider the module Γ_{loc}(π; x) of all local sections at x.

For a section φ ∈ Γ_{loc}(π; x) satisfying φ(x) = θ ∈ E, consider its graph Γ_{φ} ⊂ E and all sections φ′ ∈ Γ_{loc}(π; x) such that

(a) φ(x) = φ′(x);

(b) the graph Γ_{φ′} is tangent to Γ_{φ} with order k at θ. Conditions (a) and (b) determine equivalence relation ∼_{x}^{k} on Γ_{loc}(π; x) and we denote the equivalence class of φ by [φ]_{x}^{k}. The quotient set Γ_{loc}(π; x)/∼_{x}^{k} becomes an ℝ-vector space, if we put

[φ]_{x}^{k} + [ψ]_{x}^{k} = [φ + ψ]_{x}^{k}, a[φ]_{x}^{k} = [aφ]_{x}^{k}, φ, ψ ∈ Γ_{loc}(π; x), a ∈ ℝ,

while the natural projection Γ_{loc}(π; x) → Γ_{loc}(π; x)/ ∼_{x}^{k} becomes a linear map. We denote this quotient space by J_{x}^{k}(π). Obviously, J_{x}^{0}(π) coincides with E_{x} = π^{−1}(x).

The tangency class [φ]_{x}^{k} is completely determined by the point x and partial derivatives at x of the section φ up to order k. From here it follows that J_{x}^{k}(π) is finite-dimensional with

dim J_{x}^{k}(π) = m ∑_{i=0}^{k} \binom{n + i − 1}{n − 1} = m\binom{n + k}{k}.  (3.1)

Definition 3.1. The element [φ]_{x}^{k} ∈ J_{x}^{k}(π) is called the k-jet of the section φ ∈ Γ_{loc}(π; x) at the point x.

The k-jet of φ at x can be identified with the k-th order Taylor expansion of the section φ. From the definition it follows that it is independent of coordinate choice.

Consider now the set

J^{k}(π) = ⋃_{x∈M} J_{x}^{k}(π)  (3.2)

and introduce a smooth manifold structure on J^{k}(π) in the following way. Let {U_{α}}_{α} be an atlas in M such that the bundle π becomes trivial over each U_{α}, i.e., π^{−1}(U_{α}) ≃ U_{α} × F, where F is the “typical fiber”. Choose a basis e_{1}^{α}, …, e_{m}^{α} of local sections of π over U_{α}. Then any section of π |_{U_{α}} is representable in the form φ = u^{1}e_{1}^{α} + ⋯ + u^{m}e_{m}^{α} and the functions x_{1}, …, x_{n}, u^{1}, …, u^{m}, where x_{1}, …, x_{n} are local coordinates in U_{α}, constitute a local coordinate system in π^{−1}(U_{α}). Let us define the functions u_{σ}^{j}: ⋃_{x∈U_{α}} J_{x}^{k}(π) → ℝ, where σ = i_{1} … i_{r}, |σ| = r ≤ k, by

u_{σ}^{j}([φ]_{x}^{k}) = \frac{∂^{|σ|}u^{j}}{∂x_{i_{1}} ⋯ ∂x_{i_{r}}} |_{x} .  (3.3)

Then these functions, together with local coordinates x_{1}, …, x_{n}, determine the map f_{α}: ⋃_{x∈U_{α}} J_{x}^{k}(π) → U_{α} × ℝ^{N}, where N is the number defined by (3.1) on the page before. Due to computation rules for partial derivatives under coordinate transformations, the map

(f_{α} ∘ f_{β}^{−1}) |_{U_{α}∩U_{β}} : (U_{α} ∩ U_{β}) × ℝ^{N} → (U_{α} ∩ U_{β}) × ℝ^{N}

is a diffeomorphism preserving the natural projection (U_{α} ∩ U_{β}) × ℝ^{N} → (U_{α} ∩ U_{β}). Thus we have proved the following result:

Proposition 3.1. The set J^{k}(π) defined by (3.2) is a smooth manifold while the projection π_{k}: J^{k}(π) → M, π_{k}: [φ]_{x}^{k} ↦ x, is a smooth vector bundle.

Definition 3.2. Let π: E → M be a smooth vector bundle, dim M = n, dim E = n + m.

(1) The manifold J^{k}(π) is called the manifold of k-jets for π;

(2) The bundle π_{k}: J^{k}(π) → M is called the bundle of k-jets for π;

(3) The above constructed local coordinates {x_{i}, u_{σ}^{j}}, i = 1, …, n, j = 1, …, m, |σ| ≤ k, are called the special coordinate system on J^{k}(π) associated to the trivialization {U_{α}}_{α} of the bundle π.

Obviously, the bundle π_{0} coincides with π.

Since tangency of two manifolds with order k implies tangency with less order, there exists a map

π_{k,l}: J^{k}(π) → J^{l}(π), [φ]_{x}^{k} ↦ [φ]_{x}^{l}, k ≥ l,

which is a smooth fiber bundle. If k ≥ l ≥ s, then obviously

π_{l,s} ∘ π_{k,l} = π_{k,s}, π_{l} ∘ π_{k,l} = π_{k}.  (3.4)

On the other hand, for any section φ ∈ Γ(π) (or ∈ Γ_{loc}(π; x)) we can define the map j_{k}(φ): M → J^{k}(π) by setting j_{k}(φ)(x) = [φ]_{x}^{k}. Obviously, j_{k}(φ) ∈ Γ(π_{k}) (respectively, j_{k}(φ) ∈ Γ_{loc}(π_{k}; x)).

Definition 3.3. The section j_{k}(φ) is called the k-jet of the section φ. The correspondence j_{k}: Γ(π) → Γ(π_{k}) is called the k-jet operator.

From the definition it follows that

π_{k,l} ∘ j_{k}(φ) = j_{l}(φ), j_{0}(φ) = φ, k ≥ l,  (3.5)

for any φ ∈ Γ(π).

Let φ, ψ ∈ Γ(π) be two sections, x ∈ M and φ(x) = ψ(x) = θ ∈ E. It is a tautology to say that the manifolds Γ_{φ} and Γ_{ψ} are tangent to each other with order k +l at θ or that the manifolds Γ_{j_{k}(φ)}, Γ_{j_{k}(ψ)} ⊂ J^{k}(π) are tangent with order l at the point θ_{k} = j_{k}(φ)(x) = j_{k}(ψ)(x).

Definition 3.4. Let θ_{k} ∈ J^{k}(π). An R-plane at θ_{k} is an n-dimensional plane tangent to a manifold of the form Γ_{j_{k}(φ)} such that [φ]_{x}^{k} = θ_{k}.

Immediately from definitions we obtain the following result.

Proposition 3.2. Consider a point θ_{k} ∈ J^{k}(π). Then the fiber of the bundle π_{k+1,k}: J^{k+1}(π) → J^{k}(π) over θ_{k} coincides with the set of all R-planes at θ_{k}.

For θ_{k+1} ∈ J^{k+1}(π), we shall denote the corresponding R-plane at θ_{k} = π_{k+1,k}(θ_{k+1}) by L_{θ_{k+1}} ⊂ T_{θ_{k}}(J^{k}(π)).

3.2. Nonlinear differential operators. Let us consider now the algebra of smooth functions on J^{k}(π) and denote it by F_{k} = F_{k}(π). Take another vector bundle π′: E′ → M and consider the pull-back π_{k}^{∗}(π′). Then the set of sections of π_{k}^{∗}(π′) is a module over F_{k}(π) and we denote this module by F_{k}(π, π′). In particular, F_{k}(π) = F_{k}(π, 1_{M}), where 1_{M} is the trivial one-dimensional bundle over M.

The surjections π_{k,l} and π_{k} for all k ≥ l ≥ 0 generate the natural embeddings ν_{k,l} = π_{k,l}^{∗}: F_{l}(π, π′) → F_{k}(π, π′) and ν_{k} = π_{k}^{∗}: Γ(π′) → F_{k}(π, π′). Due to (3.4) on the facing page, we have the equalities

ν_{k,l} ∘ ν_{l,s} = ν_{k,s}, ν_{k,l} ∘ ν_{l} = ν_{k}, k ≥ l ≥ s.  (3.6)

Identifying F_{l}(π, π′) with its image in F_{k}(π, π′) under ν_{k,l}, we can consider F_{k}(π, π′) as a filtered module,

Γ(π′) ↪ F_{0}(π, π′) ↪ ⋯ ↪ F_{k−1}(π, π′) ↪ F_{k}(π, π′),  (3.7)

over the filtered algebra C^{∞}(M) ↪ F_{0} ↪ ⋯ ↪ F_{k−1} ↪ F_{k} with the embeddings F_{k} · F_{l}(π, π′) ⊂ F_{max(k , l)}(π, π′). Let F ∈ F_{k}(π, π′). Then we have the correspondence

Δ = Δ_{F} : Γ(π) → Γ(π′), Δ(φ) = j_{k}(φ)^{∗}(F), φ ∈ Γ(π).  (3.8)

Definition 3.5. A correspondence Δ of the form (3.8) is called a (nonlinear) differential operator of order ≤ k acting from the bundle π to the bundle π′. In particular, when Δ(aφ + bψ) = aΔ(φ) + bΔ(ψ), a, b ∈ ℝ, the operator Δ is said to be linear.

Example 3.1. Let us show that the k-jet operator j_{k}: Γ(π) → Γ(π_{k}) (Definition 3.3 on page 36) is differential. To do this, recall that the total space of the pull-back π_{k}^{∗}(π_{k}) consists of points (θ_{k}, θ′_{k}) ∈ J^{k}(π) × J^{k}(π) such that π_{k}(θ_{k}) = π_{k}(θ′_{k}). Consequently, we may define the diagonal section ρ_{k} ∈ F_{k}(π, π_{k}) of the bundle π_{k}^{∗}(π_{k}) by setting ρ_{k}(θ_{k}) = θ_{k}. Obviously, j_{k} = Δ_{ρ_{k}}, i.e.,

j_{k}(φ)^{∗}(ρ_{k}) = j_{k}(φ), φ ∈ Γ(π).

The operator j_{k} is linear.

Example 3.2. Let τ^{∗}: T^{∗}M → M be the cotangent bundle of M and τ_{p}^{∗}: ⋀^{p} T^{∗}M → M be its p-th exterior power. Then the de Rham differential d is a first order linear differential operator acting from τ_{p}^{∗} to τ_{p+1}^{∗}, p ≥ 0.

Let us prove now that composition of nonlinear differential operators is a differential operator again. Let Δ: Γ(π) → Γ(π′) be a differential operator of order ≤ k. For any θ_{k} = [φ]_{x}^{k} ∈ J^{k}(π), set

Φ_{Δ}(θ_{k}) = [Δ(φ)]_{x}^{0} = (Δ(φ))(x).  (3.9)

Evidently, the map Φ_{Δ} is a morphism of fiber bundles (but not of vector bundles!), i.e., π′ ∘ Φ_{Δ} = π_{k}.

Definition 3.6. The map Φ_{Δ} is called the representative morphism of the operator Δ.

For example, for Δ = j_{k} we have Φ_{j_{k}} = id_{J^{k}(π)}. Note that there exists a one-to-one correspondence between nonlinear differential operators and their representative morphisms: one can easily see it just by inverting equality (3.9). In fact, if Φ: J^{k}(π) → E′ is a morphism of π to π′, a section φ ∈ F(π, π′) can be defined by setting φ(θ_{k}) = (θ_{k}, Φ(θ_{k})) ∈ J^{k}(π) × E′. Then, obviously, Φ is the representative morphism for Δ = Δ_{φ}.

Definition 3.7. Let Δ: Γ(π) → Γ(π′) be a k-th order differential operator. Its l-th prolongation is the composition Δ^{(l)} = j_{l} ∘ Δ: Γ(π) → Γ(π_{l}).

Lemma 3.3. For any k-th order differential operator Δ, its l-th prolongation is a (k + l)-th order operator.

Proof. In fact, for any θ_{k+l} = [φ]_{x}^{k+l} ∈ J^{k+l}(π) set Φ_{Δ}^{(l)}(θ_{k+l}) = [Δ(φ)]_{x}^{l} ∈ J^{l}(π). Then the operator, for which the morphism Φ_{Δ}^{(l)} is representative, coincides with Δ^{(l)}.

Corollary 3.4. The composition Δ′ ∘ Δ of two nonlinear differential operators Δ: Γ(π) → Γ(π′) and Δ′: Γ(π′) → Γ(π′′) of orders ≤ k and ≤ k′ respectively is a (k + k′)-th order differential operator.

Proof. Let Φ_{Δ}^{(k′)} : J^{k+k′}(π) → J^{k′}(π′) be the representative morphism for Δ^{(k′)}. Then the operator □, for which the composition Φ_{Δ′} ∘ Φ_{Δ}^{(k′)} is the representative morphism, coincides with Δ′ ∘ Δ.

The following obvious proposition describes main properties of prolongations and representative morphisms.

Proposition 3.5. Let Δ: Γ(π) → Γ(π′) and Δ′: Γ(π′) → Γ(π′′) be two differential operators of orders k and k′ respectively. Then:

(1) Φ_{Δ′∘Δ} = Φ_{Δ′} ∘ Φ_{Δ}^{(k′)},

(2) Φ_{Δ}^{(l)} ∘ j_{k+l}(φ) = Δ^{(l)}(φ) for any φ ∈ Γ(π) and l ≥ 0,

(3) π_{l,l′} ∘ Φ_{Δ}^{(l)} = Φ_{Δ}^{(l′)} ∘ π_{k+l,k+l′}, i.e., the diagram

\begin{matrix}J^{k+l}(π) & ⟶^{Φ_{Δ}^{(l)}} & J^{l}(π′) \\ ^{π_{k+l,k+l′}} ↓ & & ↓^{π′_{l,l′}} \\ J^{k+l′}(π) & ⟶^{Φ_{Δ}^{(l′)}} & J^{l′}(π′) \end{matrix}  (3.10)

is commutative for all l ≥ l′ ≥ 0.

3.3. Infinite jets. We now pass to infinite limit in all previous constructions.

Definition 3.8. The space of infinite jets J^{∞}(π) of the fiber bundle π: E → M is the inverse limit of the sequence

⋯ ⟶ J^{k+1}(π) ⟶^{π_{k+1,k}} J^{k}(π) ⟶ ⋯ ⟶ J^{1}(π) ⟶^{π_{1,0}} E ⟶^{π} M,

i.e., J^{∞}(π) = proj lim_{{π_{k,l}}} J^{k}(π).

Thus a point θ of J^{∞}(π) is a sequence of points {θ_{k}}_{k≥0}, θ_{k} ∈ J^{k}(π), such that π_{k,l}(θ_{k}) = θ_{l}, k ≥ l. Points of J^{∞}(π) can be understood as mdimensional formal series and can be represented in the form θ = [φ]_{x}^{∞}, φ ∈ Γ_{loc}(π).

A special coordinate system associated to a trivialization {U_{α}}_{α} is given by the functions x_{1}, …, x_{n}, …, u_{σ}^{j}, . . ..

A tangent vector to J^{∞}(π) at a point θ is defined as a system of vectors {w, v_{k}}_{k≥0} tangent to M and to J^{k}(π) respectively such that (π_{k})_{∗}v_{k} = w, (π_{k,l})_{∗}v_{k} = v_{l} for all k ≥ l ≥ 0.

A smooth bundle ξ over J^{∞}(π) is a system of bundles η: Q → M, ξ_{k}: P_{k} → J^{k}(π) together with smooth maps Ψ_{k}: P_{k} → Q, Ψ_{k,l}: P_{k} → P_{l}, k ≥ l ≥ 0, such that

Ψ_{l} ∘ Ψ_{k,l} = Ψ_{k}, Ψ_{k,l} ∘ Ψ_{l,s} = Ψ_{k,s}, k ≥ l ≥ s ≥ 0

For example, if η: Q → M is a bundle, then the pull-backs π_{k}^{∗}(η): π_{k}^{∗}(Q) → J^{k}(π) together with natural projections π_{k}^{∗}(Q) → π_{l}^{∗}(Q) and π_{k}^{∗}(Q) → Q form a bundle over J^{∞}(π). We say that ξ is a vector bundle over J^{∞}(π), if η and all ξ_{k} are vector bundles while the maps Ψ_{k} and Ψ_{k,l} are fiberwise linear.

A smooth map of J^{∞}(π) to J^{∞}(π′), where π: E → M, π′: E′ → M′, is defined as a system F of maps F_{−∞}: M → M′, F_{k}: J^{k}(π) → J^{k−s}(π′), k ≥ s, where s ∈ ℤ is a fixed integer called the degree of F, such that

π_{k−r,k−s−1} ∘ F_{k} = F_{k−1} ∘ π_{k,k−1}, k ≥ s + 1,

and

π_{k−s} ∘ F_{k} = F_{−∞} ∘ π_{k}, k ≥ s.

For example, if Δ: Γ(π) → Γ(π′) is a differential operator of order s, then the system of maps F_{−∞} = id_{M}, F_{k} = Φ_{Δ}^{(k−s)}, k ≥ s (see the previous subsection), is a smooth map of J^{∞}(π) to J^{∞}(π′).

A smooth function on J^{∞}(π) is an element of the direct limit F = F(π) = inj lim_{{π_{k,l}^{∗}}} F_{k}(π), where F_{k}(π) is the algebra of smooth functions on J^{k}(π). Thus, a smooth function on J^{∞}(π) is a function on J^{k}(π) for some finite but an arbitrary k. The set F = F(π) of such functions is identified with ⋃_{k=0}^{∞} F_{k}(π) and forms a commutative filtered algebra. Using duality between smooth manifolds and algebras of smooth functions on these manifolds, we deal in what follows with the algebra F(π) rather than with the manifold J^{∞}(π) itself.

From this point of view, a vector field on J^{∞}(π) is a filtered derivation of F(π), i.e., an ℝ-linear map X: F(π) → F(π) such that

X(fg) = fX(g) + gX(f), f, g ∈ F(π), X(F_{k}(π)) ⊂ F_{k+l}(π)

for all k and some l = l(X). The latter is called the filtration degree of the field X. The set of all vector fields is a filtered Lie algebra over ℝ with respect to commutator [X, Y] and is denoted by D(π) = ⋃_{l≥0} D^{(l)}(π).

Differential forms of degree i on J^{∞}(π) are defined as elements of the filtered F(π)-module Λ^{i} = Λ^{i}(π) = ⋃_{k≥0} Λ^{i}(π_{k}), where Λ^{i}(π_{k}) = Λ^{i}(J^{k}(π)) and the module Λ^{i}(π_{k}) is considered to be embedded into Λ^{i}(π_{k+1}) by the map π_{k+1,k}^{∗}. Defined in such a way, these forms possess all basic properties of differential forms on finite-dimensional manifolds. Let us mention the most important ones:

(1) The module Λ^{i}(π) is the i-th exterior power of Λ^{1}(π), Λ^{i}(π) = ⋀^{i} Λ^{1}(π). Respectively, the operation of wedge product ∧: Λ^{p}(π) ⊗ Λ^{q}(π) → Λ^{p+q}(π) is defined and Λ^{∗}(π) = ⨁_{i≥0} Λ^{i}(π) becomes a supercommutative ℤ-graded algebra.

(2) The module D(π) is dual to Λ^{1}(π), i.e.,

D(π) = Hom_{F(π)}^{ϕ}(Λ^{1}(π), F(π)),  (3.11)

where Hom_{F(π)}^{ϕ}(·, ·) denotes the module of filtered homomorphisms over F(π). Moreover, equality (3.11) is established in the following way: there is a derivation d: F(π) → Λ^{1}(π) (the de Rham differential on J^{∞}(π)) such that for any vector field X there exists a uniquely defined filtered homomorphism f_{X} satisfying f_{X} ∘ d = X.

(3) The operator d is extended up to maps d: Λ^{i}(π) → Λ^{i+1}(π) in such a way that the sequence

0 ⟶ F(π) ⟶^{d} Λ^{1}(π) ⟶ ⋯ ⟶ Λ^{i}(π) ⟶^{d} Λ^{i+1}(π) ⟶ ⋯

becomes a complex, i.e., d∘d = 0. This complex is called the de Rham complex on J^{∞}(π). The latter is a derivation of the superalgebra Λ^{∗}(π).

Using algebraic techniques (see Section 1), we can introduce the notions of inner product and Lie derivative and to prove their basic properties (cf. Proposition 1.14 on page 15). We can also define linear differential operators over J^{∞}(π) as follows. Let P and Q be two filtered F(π)-modules and Δ ∈ Hom_{F(π)}^{ϕ}(P, Q). Then Δ is called a linear differential operator of order ≤ k acting from P to Q, if

(δ_{f_{0}} ∘ δ_{f_{1}} ∘ ⋯ ∘ δ_{f_{k}})Δ = 0

for all f_{0}, …, f_{k} ∈ F(π), where, as in Section 1, (δ_{f}Δ)p = Δ(fp) − fΔ(p). We write k = ord(Δ).

Due to existence of filtrations in the algebra F(π), as well as in modules P and Q, one can define differential operators of infinite order acting from P to Q. Namely, let P = {P_{l}}, Q = {Q_{l}}, P_{l} ⊂ P_{l+1}, Q_{l} ⊂ Q_{l+1}, P_{l} and Q_{l} being F_{l}(π)-modules. Let Δ ∈ Hom_{F(π)}^{ϕ}(P, Q) and s be filtration of Δ, i.e., Δ(P_{l}) ⊂ Q_{l+s}. We can always assume that s ≥ 0. Suppose now that Δ_{l} = Δ |_{P_{l}} : P_{l} → Q_{l} is a linear differential operator of order o_{l} over F_{l}(π) for any l. Then we say that Δ is a linear differential operator of order growth o_{l}. In particular, if o_{l} = αl + β, α, β ∈ ℝ, we say that Δ is of constant growth α.

Distributions. Let θ ∈ J^{∞}(π). The tangent plane to J^{∞}(π) at the point θ is the set of all tangent vectors to J^{∞}(π) at this point (see above). Denote such a plane by T_{θ} = T_{θ}(J^{∞}(π)). Let θ = {x, θ_{k}}, x ∈ M, θ_{k} ∈ J^{k}(π) and v = {w, v_{k}}, v′ = {w′, v′_{k}} ∈ T_{θ}. Then the linear combination λv + μv′ = {λw + μw′, λv_{k} + μv′_{k}} is again an element of T_{θ} and thus T_{θ} is a vector space. A correspondence T : θ ↦ T_{θ} ⊂ T_{θ}, where T_{θ} is a linear subspace, is called a distribution on J^{∞}(π). Denote by T D(π) ⊂ D(π) the submodule of vector fields lying in T , i.e., a vector field X belongs to T D(π) if and only if X_{θ} ∈ T_{θ} for all θ ∈ J^{∞}(π). We say that the distribution T is integrable, if it satisfies the formal Frobenius condition: for any vector fields X, Y ∈ T D(π) their commutator lies in T D(π) as well, or [T D(π), T D(π)] ⊂ T D(π).

This condition can expressed in a dual way as follows. Let us set

T^{1}Λ(π) = { ω ∈ Λ^{1}(π) | i_{X}ω = 0, X ∈ T D(π) }

and consider the ideal T Λ^{∗}(π) generated in Λ^{∗}(π) by T^{1}Λ(π). Then the distribution T is integrable if and only if the ideal T Λ^{∗}(π) is differentially closed: d(T Λ^{∗}(π)) ⊂ T Λ^{∗}(π).

Finally, we say that a submanifold N ⊂ J^{∞}(π) is an integral manifold of T , if T_{θ}N ⊂ T_{θ} for any point θ ∈ N. An integral manifold N is called locally maximal at a point θ ∈ N, if there no neighborhood U ⊂ N of θ is embedded to other integral manifold N′ such that dim N ≤ dim N′.

3.4. Nonlinear equations and their solutions. Let π: E → M be a vector bundle.

Definition 3.9. A submanifold E ⊂ J^{k}(π) is called a (nonlinear) differential equation of order k in the bundle π. We say that E is a linear equation, if E ∩ π_{x}^{−1}(x) is a linear subspace in π_{x}^{−1}(x) for all x ∈ M. In other words, E is a linear subbundle in the bundle π_{k}.

We shall always assume that E is projected surjectively to E under π_{k,0}.

Definition 3.10. A (local) section f of the bundle π is called a (local) solution of the equation E, if its graph lies in E: j_{k}(f)(M) ⊂ E.

We say that the equation E is determined, if codim E = dim π, that it is overdetermined, if codim E > dim π, and that it is underdetermined, if codim E < dim π.

Obviously, in a special coordinate system these definitions coincide with “usual” ones.

One of the ways to represent differential equations is as follows. Let π′: ℝ^{r} × Ū → Ū be the trivial r-dimensional bundle. Then the set of functions (F^{1}, …, F^{r}) can be understood as a section φ of the pull-back (π_{k} |_{U})^{∗} (π′), or as a nonlinear operator Δ = Δ_{φ} defined in U, while the equation E is characterized by the condition

E ∩ U = { θ_{k} ∈ U | φ(θ_{k}) = 0 }.  (3.12)

More generally, any equation E ⊂ J^{k}(π) can be represented in the form similar to (3.12). Namely, for any equation E there exists a fiber bundle π′: E′ → M and a section φ ∈ F_{k}(π, π) such that E coincides with the set of zeroes for φ: E = {φ = 0}. In this case we say that E is associated to the operator Δ = Δ_{φ}: Γ(π) → Γ(π′) and use the notation E = E_{Δ}.

Example 3.3. Let π = τ_{p}^{∗}: ⋀^{p} T^{∗}M → M, π′ = τ_{p+1}^{∗}: ⋀^{p+1} T^{∗}M → M and d: Γ(π) = Λ^{p}(M) → Γ(π′) = Λ^{p+1}(M) be the de Rham differential (see Example 3.2 on page 38). Thus we obtain a first-order equation E_{d} in the bundle τ_{p}^{∗}. Consider the case p = 1, n ≥ 2 and choose local coordinates x_{1}, …, x_{n} in M. Then any form ω ∈ Λ^{1}(M) is represented as ω = u^{1}dx_{1} + ⋯ + u^{n}dx_{n} and we have E_{d} = { u_{i}^{j} = u_{j}^{i} | i < j }. This equation is underdetermined when n = 2, determined for n = 3 and overdetermined for n > 3.

Example 3.4. Consider an arbitrary vector bundle π: E → M and a differential form ω ∈ Λ^{p}(J^{k}(π)), p ≤ dim M. Then the condition j_{k}(φ)^{∗}(ω) = 0, φ ∈ Γ(π), determines a (k + 1)-st order equation E_{ω} in the bundle π. Consider the case p = dim M = 2, k = 1 and choose a special coordinate system x, y, u, u_{x}, u_{y} in J^{1}(π). Let φ = φ(x, y) be a local section and

ω = Adu_{x} ∧ du_{y} + (B_{1}du_{x} + B_{2}du_{y}) ∧ du + du_{x} ∧ (B_{11}dx + B_{12}dy) + du_{y} ∧ (B_{21}dx + B_{22}dy) + du ∧ (C_{1}dx + C_{2}dy) + Ddx ∧ dy,

where A, B_{i}, B_{ij}, C_{i}, D are functions of x, y, u, u_{x}, u_{y}. Then we have

j_{1}(φ)^{∗}ω = (A^{φ}(φ_{xx}φ_{yy} − φ_{xy}^{2}) + (φ_{y}B_{1}^{φ} + B_{12}^{φ})φ_{xx}

− (φ_{x}B_{2}^{φ} + B_{12}^{φ})φ_{yy} + (φ_{y}B_{2}^{φ} − φ_{x}B_{1}^{φ} + B_{22}^{φ} − B_{11}^{φ})φ_{xy}

+ φ_{x}C_{2}^{φ} − φ_{y}C_{1}^{φ} + D^{φ}))dx ∧ dy,

where F^{φ} = j_{1}(φ)^{∗}F for any F ∈ F_{1}(π). Hence, the equation E_{ω} is of the form

a(u_{xx}u_{yy} − u_{xy}^{2}) + b_{11}u_{xx} + b_{12}u_{xy} + b_{22}u_{yy} + c = 0,  (3.13)

where a = A, b_{11} = u_{y}B_{1} + B_{12}, b_{12} = u_{y}B_{2} − u_{x}B_{1} + B_{22} − B_{11}, b_{22} = u_{x}B_{2} + B_{12}, c = u_{x}C_{2} − u_{y}C_{1} + D are functions on J^{1}(π). Equation (3.13) is the so-called two-dimensional Monge–Ampere equation and obviously any such an equation can be represented as E_{ω} for some ω ∈ Λ^{1}(J^{1}(π)) (see [36] for more details).

Example 3.5. Consider again a bundle π: E → M and a section ∇: E → J^{1}(π) of the bundle π_{1,0} : J^{1}(π) → E. Then the graph E_{∇} = ∇(E) ⊂ J^{1}(π) is a first-order equation in the bundle π. Let θ_{1} ∈ E_{∇}. Then, due to Proposition 3.2 on page 37, the point θ_{1} is identified with the pair (θ_{0}, L_{θ_{1}}), where θ_{0} = π_{1,0}(θ_{1}) ∈ E, while L_{θ_{1}} is the R-plane at θ_{0} corresponding to θ_{1}. Hence, the section ∇ (or the equation E_{∇}) may be understood as a distribution of horizontal (i.e., nondegenerately projected to T_{x}M under (π_{k})_{∗}, where x = π_{k}(θ_{k})) n-dimensional planes on E: T_{∇}: E ∋ θ ↦ θ_{1} = L_{∇(θ)}. In other words, ∇ is a connection in the bundle π. A solution of the equation E_{∇}, by definition, is a section φ ∈ Γ(π) such that j_{1}(φ)(M) ⊂ ∇(E). It means that at any point θ = φ(x) ∈ φ(M) the plane T_{∇}(θ) is tangent to the graph of the section φ. Thus, solutions of E_{∇} coincide with integral manifolds of T_{∇}.

In a local coordinate system (x_{1}, …, x_{n}, u^{1}, …, u^{m}, …, u_{i}^{j}, …), i = 1, …, n, j = 1, …, m, the equation E_{∇} is represented as

u_{i}^{j} = ∇_{i}^{j}(x_{1}, …, x_{n}, u^{1}, …, u^{m}), i = 1, …, n, j = 1, …, m,  (3.14)

∇_{i}^{j} being smooth functions.

Example 3.6. As we saw in the previous example, to solve the equation E_{∇} is the same as to find integral n-dimensional manifolds of the distribution T_{∇}. Hence, the former to be solvable, the latter is to satisfy the Frobenius theorem. Thus, for solvable E_{∇} we obtain conditions on the section ∇ ∈ Γ(π_{1,0}). Let write down these conditions in local coordinates.

Using representation (3.14) on the preceding page, note that T_{∇} is given by 1-forms

ω^{j} = du^{j} − ∑_{i=1}^{n} ∇_{i}^{j}dx_{i}, j = 1, …, m.

Hence, the integrability conditions may be expressed as

dω^{j} = ∑_{i=1}^{m} ρ_{i}^{j} ∧ ω_{i}, j = 1, …, m,

for some 1-forms ρ_{i}^{i}. After elementary computations, we obtain that the functions ∇_{i}^{j} must satisfy the following relations:

\frac{∂∇_{α}^{j}}{∂x_{β}} + ∑_{γ=1}^{m} ∇_{α}^{γ} \frac{∂∇_{β}^{j}}{∂u^{γ}} = \frac{∂∇_{β}^{j}}{∂x_{α}} + ∑_{γ=1}^{m} ∇_{β}^{γ} \frac{∂∇_{α}^{j}}{∂u^{γ}}  (3.15)

for all j = 1, …, m, 1 ≤ α < β ≤ m. Thus we got a naturally constructed first-order equation I(π) ⊂ J^{1}(π_{1,0}), whose solutions are horizontal n-dimensional distributions in E = J^{0}(π).

3.5. Cartan distribution on J^{k}(π). We shall now introduce a very important structure on J^{k}(π) responsible for “individuality” of these manifolds.

Definition 3.11. Let π: E → M be a vector bundle. Consider a point θ_{k} ∈ J^{k}(π) and the span C_{θ_{k}}^{k} ⊂ T_{θ_{k}}(J^{k}(π)) of all R-planes at the point θ_{k}.

(1) The correspondence C^{k} = C^{k}(π): θ_{k} ↦ C_{θ_{k}}^{k} is called the Cartan distribution on J^{k}(π).

(2) Let E ⊂ J^{k}(π) be a differential equation. Then the correspondence C^{k}(E): E ∋ θ_{k} ↦ C_{θ_{k}}^{k} ∩ T_{θ_{k}}E ⊂ T_{θ_{k}}E is called the Cartan distribution on E. We call elements of the Cartan distributions Cartan planes.

(3) A point θ_{k} ∈ E is called regular, if the Cartan plane C_{θ_{k}}^{k}(E) is of maximal dimension. We say that E is a regular equation, if all its points are regular.

In what follows, we deal with regular equations or with neighborhoods of regular points. As it can be easily seen, for any regular point there exists a neighborhood of this point all points of which are regular.

Let θ_{k} ∈ J^{k}(π) be represented in the form

θ_{k} = [φ]_{x}^{k}, φ ∈ Γ(π), x = π_{k}(θ_{k}).  (3.16)

Then, by definition, the Cartan plain C_{θ_{k}}^{k} is spanned by the vectors

j_{k}(φ)_{∗,x}(v), v ∈ T_{x}M,  (3.17)

for all φ ∈ Γ_{loc}(π) satisfying (3.16) on the preceding page.

Let x_{1}, …, x_{n}, …, u_{σ}^{j}, …, j = 1, …, m, |σ| ≤ k, be a special coordinate system in a neighborhood of θ_{k}. The vectors of the form (3.17) can be expressed as linear combinations of the vectors

\frac{∂}{∂x_{i}} + ∑_{|σ|≤k} ∑_{j=1}^{m} \frac{∂^{|σ|+1}φ^{j}}{∂x_{σ}∂x_{i}} \frac{∂}{∂u_{σ}^{j}} ,  (3.18)

where i = 1, …, n. Using this representation, we prove the following result:

Proposition 3.6. For any point θ_{k} ∈ J^{k}(π), k ≥ 1, the Cartan plane C_{θ_{k}}^{k} is of the form C_{θ_{k}}^{k} = (π_{k,k−1})_{∗}^{−1}(L_{θ_{k}}), where L_{θ_{k}} is the R-plane at the point π_{k,k−1}(θ_{k}) ∈ J^{k−1}(π) determined by the point θ_{k}.

Proof. Denote the vector (3.18) by v_{i}^{k,φ}. It is obvious that for any two sections φ and φ′ satisfying (3.16) on the facing page the difference v_{i}^{k,φ} − v_{i}^{k,φ′} is a π_{k,k−1}-vertical vector and any such a vector can be obtained in this way. On the other hand, the vectors v_{i}^{k−1,φ} do not depend on section φ satisfying (3.16) on the preceding page and form a basis in the space L_{θ_{k}}.

Remark 3.1. From the result proved it follows that the Cartan distribution on J^{k}(π) can be locally considered as generated by the vector fields

D_{i}^{(k−1)} = \frac{∂}{∂x_{i}} + ∑_{|σ|≤k−1} ∑_{j=1}^{m} u_{σi}^{j} \frac{∂}{∂u_{σ}^{j}} , V_{τ}^{s} = \frac{∂}{∂u_{τ}^{s}} , |τ| = k, s = 1, …, m. (3.19)

From here, by direct computations, it follows that [V_{τ}^{s}, D_{i}^{(k−1)}] = V_{τ−i}^{s}, where

V_{τ−i}^{s} = \{ \begin{matrix}V_{τ′}, & if τ = τ′i, \\ 0, & if τ does not contain i. \end{matrix}

But, as it follows from Proposition 3.6, vector fields V_{σ}^{j} for |σ| ≤ k do not lie in C^{k}. Thus, the Cartan distribution on J^{k}(π) is not integrable.

Introduce 1-forms in special coordinates on J^{k+1}(π):

ω_{σ}^{j} = du_{σ}^{j} − ∑_{i=1}^{n} u_{σi}^{j}dx_{i},  (3.20)

where j = 1, …, m, |σ| < k. From the representation (3.19) we immediately obtain the following important property of the forms introduced:

Proposition 3.7. The system of forms (3.20) on the preceding page annihilates the Cartan distribution on J^{k}(π), i.e., a vector field X lies in C^{k} if and only if i_{X}ω_{σ}^{j} = 0 for all j = 1, …, m, |σ| < k.

Definition 3.12. The forms (3.20) on the page before are called the Cartan forms on J^{k}(π) associated to the special coordinate system x_{i}, u_{σ}^{j}.

Note that the F_{k}(π)-submodule generated in Λ^{1}(J^{k}(π) by the forms (3.20) on the preceding page is independent of the choice of coordinates.

Definition 3.13. The F_{k}(π)-submodule generated in Λ^{1}(J^{k}(π)) by the Cartan forms is called the Cartan submodule. We denote this submodule by CΛ^{1}(J^{k}(π)).

We shall now describe maximal integral manifolds of the Cartan distribution on J^{k}(π). Let N ⊂ J^{k}(π) be an integral manifold of the Cartan distribution. Then from Proposition 3.7 it follows that the restriction of any Cartan form ω to N vanishes. Similarly, the differential dω vanishes on N. Therefore, if vector fields X, Y are tangent to N, then dω |_{N} (X, Y) = 0.

Definition 3.14. Let C_{θ_{k}}^{k} be the Cartan plane at θ ∈ J^{(}π).

(1) Two vectors v, w ∈ C_{θ_{k}}^{k} are said to be in involution, if dω_{θ_{k}}(v, w) = 0 for any ω ∈ CΛ^{1}(J^{k}(π)).

(2) A subspace W ⊂ C_{θ_{k}}^{k} is said to be involutive, if any two vectors v, w ∈ W are in involution.

(3) An involutive subspace is called maximal, if it cannot be embedded into any other involutive subspace.

Consider a point θ_{k} = [φ]_{x}^{k} ∈ J^{k}(π). Then from Proposition 3.7 it follows that the direct sum decomposition C_{θ_{k}}^{k} = T_{θ_{k}}^{v} ⊕ T_{θ_{k}}^{φ} takes place, where T_{θ_{k}}^{v} denotes the tangent plane to the fiber of the projection π_{k,k−1} passing through the point θ_{k}, while T_{θ_{k}}^{φ} is the tangent plane to the graph of j_{k}(φ). Hence, the involutiveness is sufficient to be checked for the following pairs of vectors v, w ∈ C_{θ_{k}}^{k}:

(1) v, w ∈ T_{θ_{k}}^{v};

(2) v, w ∈ T_{θ_{k}}^{φ};

(3) v ∈ T_{θ_{k}}^{v}, w ∈ T_{θ_{k}}^{φ}.

Note now that the tangent space T_{θ_{k}}^{v} is identified with the tensor product S^{k}(T_{x}^{∗})⊗E_{x}, x = π_{k}(θ_{k}) ∈ M, where T_{x}^{∗} is the fiber of the cotangent bundle to M at x, E_{x} is the fiber of the bundle π at the same point while S^{k} denotes the k-th symmetric power. Then any vector w ∈ T_{x}M determines the map δ_{w}: S^{k}(T_{x}^{∗}) ⊗ E_{x} → S^{k−1}(T_{x}^{∗}) ⊗ E_{x} by

δ_{w}(ρ_{1} · . . . · ρ_{k}) ⊗ e = ∑_{i=1}^{k} ρ_{1} · . . . · ⟨ρ_{i}, w⟩ · . . . · ρ_{k} ⊗ e,

where the dot “ · ” denotes multiplication in S^{k}(T_{x}^{∗}), ρ_{i} ∈ T_{x}^{∗}, e ∈ E_{x} while ⟨·, ·⟩ is the natural pairing between T_{x}^{∗} and T_{x}.

Proposition 3.8. Let v, w ∈ C_{θ_{k}}^{k}. Then:

(1) All pairs v, w ∈ T_{θ_{k}}^{v} are in involution.

(2) All pairs v, w ∈ T_{θ_{k}}^{φ} are in involution too. If v ∈ T_{θ_{k}}^{v} and w ∈ T_{θ_{k}}^{φ}, then they are in involution if and only if δ_{(π_{k})_{∗}(w)}v = 0.

Proof. Note first that the involutiveness conditions are sufficient to check for the Cartan forms (3.20) on page 45 only. The all three results follow from the representation (3.19) on page 45 by straightforward computations.

Let θ_{k} ∈ J^{k}(π) and F_{θ_{k}} be the fiber of the bundle π_{k,k−1} passing through the point θ_{k} while H ⊂ T_{x}M be a linear subspace. Using the linear structure, we identify the fiber F_{θ_{k}} of the bundle π_{k,k−1} with its tangent space and define the space

Ann(H) = { v ∈ F_{θ_{k}} | δ_{w}v = 0, ∀w ∈ H }.

Then, as it follows from Proposition 3.8, the following description of maximal involutive subspaces takes place:

Corollary 3.9. Let θ_{k} = [φ]_{x}^{k}, φ ∈ Γ_{loc}(π). Then any maximal involutive subspace V ⊂ C_{θ_{k}}^{k}(π) is of the form V = j_{k}(φ)_{∗}(H) ⊕ Ann(H) for some H ⊂ T_{x}M.

If V is a maximal involutive subspace, then the corresponding space H is obviously π_{k,∗}(V). We call the dimension of H the type of the maximal involutive subspace V and denote it by tp(V).

Proposition 3.10. Let V be a maximal involutive subspace. Then

dim V = m\binom{n − r + k − 1}{k} + r,

where n = dim M, m = dim π, r = tp(V).

Proof. Let us choose local coordinates in M in such a way that the vectors ∂/∂x_{1}, …, ∂/∂x_{r} form a basis in H. Then, in the corresponding special system in J^{k}(π), coordinates along Ann(H) will consist of those functions u_{σ}^{j}, |σ| = k, for which multi-index σ does not contain indexes 1, …, r.

Let N ⊂ J^{k}(π) be a maximal integral manifold of the Cartan distribution and θ_{k} ∈ N. Then the tangent plane to N at θ_{k} is a maximal involutive plane. Let its type be equal to r(θ_{k}).

Definition 3.15. The number tp(N) = max_{θ_{k}∈N} r(θ_{k}) is called the type of the

maximal integral manifold N of the Cartan distribution.

Obviously, the set g(N) = { θ_{k} ∈ N | r(θ_{k}) = tp(N) } is everywhere dense in N. We call the points θ_{k} ∈ g(N) generic. Let θ_{k} be such a point and U be its neighborhood in N consisting of generic points. Then:

(1) N′ = π_{k,k−1}(N) is an integral manifold of the Cartan distribution on J^{k−1}(π);

(2) dim(N′) = tp(N) and

(3) π_{k−1} |_{N′} : N′ → M is an immersion.

Theorem 3.11. Let N ⊂ J^{k−1}(π) be an integral manifold of the Cartan distribution on J^{k}(π) and U ⊂ N be an open domain consisting of generic points. Then

U = { θ_{k} ∈ J^{k}(π) | L_{θ_{k}} ⊃ T_{θ_{k−1}}U′ },

where θ_{k−1} = π_{k,k−1}(θ_{k}), U′ = π_{k,k−1}(U).

Proof. Let M′ = π_{k−1}(U′) ⊂ M. Denote its dimension (which equals to tp(N)) by r and choose local coordinates in M in such a way that the submanifold V′ is determined by the equations x_{r+1} = ⋯ = x_{n} = 0 in these coordinates. Then, since U′ ⊂ J^{k−1}(π) is an integral manifold and π_{k−1} |_{U′} : U′ → V′ is a diffeomorphism, in corresponding special coordinates the manifold U′ is given by the equations

u_{σ}^{j} = \{ \begin{matrix}\frac{∂^{|σ|}φ^{j}}{∂x_{σ}} , & if σ does not contain r + 1, …, n, \\ 0 & otherwise, \end{matrix}

for all j = 1, …, m, |σ| ≤ k−1 and some smooth function φ = φ(x_{1}, …, x_{r}). Hence, the tangent plane H to U′ at θ_{k−1} is spanned by the vectors of the form (3.18) on page 45 with i = 1, …, r. Consequently, a point θ_{k}, such that L_{θ_{k}} ⊃ H, is determined by the coordinates

u_{σ}^{j} = \{ \begin{matrix}\frac{∂^{|σ|}φ^{j}}{∂x_{σ}} , & if σ does not contain r + 1, …, n, \\ arbitrary real numbers & otherwise, \end{matrix}

where j = 1, …, m, |σ| ≤ k. Hence, if θ_{k}, θ′_{k} are two such points, then the vector θ_{k} −θ′_{k} lies in Ann(H), as it follows from the proof of Proposition 3.10 on the page before. As it can be easily seen, any integral manifold of the Cartan distribution projecting to U′ is contained in U, which concludes the proof.

Remark 3.2. Note that maximal integral manifolds N of type dim M are exactly graphs of jets j_{k}(φ), φ ∈ Γ_{loc}(π). On the other hand, if tp(N) = 0, then N coincides with a fiber of the projection π_{k,k−1} : J^{k}(π) → J^{k−1}(π).

3.6. Classical symmetries. Having the basic structure on J^{k}(π), we can now introduce transformations preserving this structure.

Definition 3.16. Let U, U′ ⊂ J^{k}(π) be open domains.

(1) A diffeomorphism F: U → U′ is called a Lie transformation, if it preserves the Cartan distribution, i.e., F_{∗}(C_{θ_{k}}^{k}) = C_{F (θ_{k})}^{k} for any point θ_{k} ∈ U.

Let E, E′ ⊂ J^{k}(π) be differential equations.

(2) A Lie transformation F: U → U is called a (local) equivalence, if F (U ∩ E) = U′ ∩ E′.

(3) A (local) equivalence is called a (local) symmetry, if E = E′ and U = U′.

Below we shall not distinguish between local and global versions of the concepts introduced.

Example 3.7. Consider the case J^{0}(π) = E. Then, since any n-dimen-sional horizontal plane in T_{θ}E is tangent to some section of the bundle π, the Cartan plane C_{θ}^{0} coincides with the whole space T_{θ}E. Thus the Cartan distribution is trivial in this case and any diffeomorphism of E is a Lie transformation.

Example 3.8. Since the Cartan distribution on J^{k}(π) is locally determined by the Cartan forms (see (3.20) on page 45), the condition of F to be a Lie transformation can be reformulated as

F^{∗}ω_{σ}^{j} = ∑_{α=1}^{m} ∑_{|τ|<k} λ_{σ,τ}^{j,α}ω_{τ}^{α}, j = 1, …, m, |σ| < k,  (3.21)

where λ_{σ,τ}^{j,α} are smooth functions on J^{k}(π). Equations (3.21) are the base for computations in local coordinates.

In particular, if dim π = 1 and k = 1, equations (3.21) reduce to the only condition F^{∗}ω = λω, where ω = du − ∑_{i=1}^{n} u_{i} dx_{i}. Hence, Lie transformations in this case are just contact transformations of the natural contact structure in J^{1}(π).

Let F: J^{k}(π) → J^{k}(π) be a Lie transformation. Then graphs of k-jets are taken by F to n-dimensional maximal manifolds. Let θ_{k+1} be a point of J^{k+1}(π) and represent θ_{k+1} as a pair (θ_{k}, L_{θ_{k+1}}), or, which is the same, as a class of graphs of k-jets tangent to each other at θ_{k}. Then the image F_{∗}(L_{θ_{k+1}}) will almost always be an R-plane at F (θ_{k}). Denote the corresponding point in J^{k+1}(π) by F^{(1)}(θ_{k+1}).

Definition 3.17. Let F: J^{k}(π) → J^{k}(π) be a Lie transformation. The above defined map F^{(1)} : J^{k+1}(π) → J^{k+1}(π) is called the 1-lifting of F.

The map F^{(1)} is a Lie transformation at the domain of its definition, since almost everywhere it takes graphs of (k +1)-jets to graphs of the same kind. Hence, for any l ≥ 1 we can define F^{(l)} = (F^{(l−1)})^{(1)} and call this map the l-lifting of F.

Theorem 3.12. Let π: E → M be an m-dimensional vector bundle over an n-dimensional manifold M and F: J^{k}(π) → J^{k}(π) be a Lie transformation. Then:

(1) If m > 1 and k > 0, then the map F is of the form F = G^{(k)} for some diffeomorphism G: J^{0}(π) → J^{0}(π);

(2) If m = 1 and k > 1, then the map F is of the form F = G^{(k−1)} for some contact transformation G: J^{1}(π) → J^{1}(π).

Proof. Recall that fibers of the projection π_{k,k−1} : J^{k}(π) → J^{k−1}(π) for k ≥ 1 are maximal integral manifolds of the Cartan distribution of type 0 (see Remark 3.2 on page 48). Further, from Proposition 3.10 on page 47 it follows in the cases m > 1, k > 0 and m = 1, k > 1 that they are integral manifolds of maximal dimension, provided n > 1. Therefore, the map F is π_{k,ϵ}-fiberwise, where ϵ = 0 for m > 1 and ϵ = 1 for m = 1.

Thus there exists a map G: J^{ϵ}(π) → J^{ϵ}(π) such that π_{k,ϵ}∘F = G∘π_{k,ϵ} and G is a Lie transformation in an obvious way. Let us show that F = G^{(k−ϵ)}. To do this, note first that in fact, by the same reasons, the transformation F generates a series of Lie transformations G_{l}: J^{l}(π) → J^{l}(π), l = ϵ, …, k, satisfying π_{l,l−1} ∘ G_{l} = G_{l−1} ∘ π_{l,l−1} and G_{k} = F, G_{ϵ} = G. Let us compare the maps F and G_{k−1}^{(1)}.

From Proposition 3.6 on page 45 and the definition of Lie transformations we obtain

F_{∗}((π_{k,k−1})_{∗}^{−1}(L_{θ_{k}})) = F_{∗}(C_{θ_{k}}^{k}) = C_{F (θ_{k})} = (π_{k,k−1})_{∗}^{−1}(L_{F (θ_{k})})

for any θ_{k} ∈ J^{k}(π). But

F_{∗}((π_{k,k−1})_{∗}^{−1}(L_{θ_{k}})) = (π_{k,k−1})_{∗}^{−1}(G_{k−1,∗}(L_{θ_{k}}))

and consequently G_{k−1,∗}(L_{θ_{k}}) = L_{F (θ_{k})}. Hence, by the definition of 1-lifting we have F = G_{k−1}^{(1)}. Using this fact as a base of elementary induction, we obtain the result of the theorem for dim M > 1.

Consider the case n = 1, m = 1 now. Since all maximal integral manifolds are one-dimensional in this case, it should be treated in a special way. Denote by V the distribution consisting of vector fields tangent to the fibers of the projection π_{k,k−1}. We must show that

F_{∗}V = V  (3.22)

for any Lie transformation F, which is equivalent to F being π_{k,k−1}- fiberwise.

Let us prove (3.22) on the facing page. To do it, consider an arbitrary distribution P on a manifold N and introduce the notation

PD = { X ∈ D(N) | X lies in P }  (3.23)

and

D_{P} = { X ∈ D(N) | [X, Y] ∈ P, ∀Y ∈ PD }.  (3.24)

Then one can show (using coordinate representation, for example) that DV = DC^{k} ∩ D_{[DC^{k},DC^{k}]} for k ≥ 2. But Lie transformations preserve the distributions at the right-hand side of the last equality and consequently preserve DV.

Definition 3.18. Let π: E → M be a vector bundle and E ⊂ J^{k}(π) be a k-th order differential equation.

(1) A vector field X on J^{k}(π) is called a Lie field, if the corresponding one-parameter group consists of Lie transformations.

(2) A Lie field is called an infinitesimal symmetry of the equation E, if it is tangent to E.

Since in the sequel we shall deal with infinitesimal symmetries only, we shall call them just symmetries. By definition, one-parameter groups of transformations corresponding to symmetries preserve generalized solutions^{6}.

Let X be a Lie field on J^{k}(π) and F_{t}: J^{k}(π) → J^{k}(π) be its one-parameter group. Then we can construct the l-lifting F_{t}^{(l)} : J^{k+l}(π) → J^{k+l}(π) and the corresponding Lie field X^{(l)} on J^{k+l}(π). This field is called the l-lifting of the field X. As we shall see a bit later, liftings of Lie fields are defined globally and can be described explicitly. An immediate consequence of the definition and of Theorem 3.12 on the preceding page is

Theorem 3.13. Let π: E → M be an m-dimensional vector bundle over an n-dimensional manifold M and X be a Lie field on J^{k}(π). Then:

(1) If m > 1 and k > 0, the field X is of the form X = Y^{(k)} for some vector field Y on J^{0}(π);

(2) If m = 1 and k > 1, the field X is of the form X = Y^{(k−1)} for some contact vector field Y on J^{1}(π).

To finish this subsection, we describe coordinate expressions for Lie fields. Let (x_{1}, …, x_{n}, …, u_{σ}^{j}, …) be a special coordinate system in J^{k}(π). Then from (3.21) on page 49 it follows that

X = ∑_{i=1}^{n} X_{i} \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} ∑_{|σ|≤k} X_{σ}^{j} \frac{∂}{∂u_{σ}^{j}}

6A generalized solution of an equation E is a maximal integral manifold N ⊂ E of the Cartan distribution on E; see [35].

is a Lie field if and only if

X_{σi}^{j} = D_{i}(X_{σ}^{j}) − ∑_{α=1}^{n} u_{σα}^{j}D_{i}(X_{α}),  (3.25)

where

D_{i} = \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} ∑_{|σ|≥0} u_{σi}^{j} \frac{∂}{∂u_{σ}^{j}}  (3.26)

are the so-called total derivatives.

Exercise 3.1. It is easily seen that the operators (3.26) do not preserve the algebras F_{k}: they are derivations acting from F_{k} to F_{k+1}. Prove that nevertheless for any contact field on J^{1}(π), dim π = 1, or for an arbitrary vector field on J^{0}(π) (regardless of the dimension of π) the formulas above determine a vector field on J^{k}(π).

Recall now that a contact field X on J^{1}(π) is completely determined by its generating function f = i_{X}ω, where ω = du − ∑_{i} u_{i} dx_{i} is the Cartan (contact) form on J^{1}(π). The contact field corresponding to f ∈ F_{1}(π) is denoted by X_{f} and is given by the expression

\begin{aligned}X_{f} = − ∑_{i=1}^{n} \frac{∂f}{∂u_{1_{i}}} \frac{∂}{∂x_{i}} + (f− & ∑_{i=1}^{n} u_{i} \frac{∂f}{∂u_{i}} )\frac{∂}{∂u} \\ & + ∑_{i=1}^{n} (\frac{∂f}{∂x_{i}} + u_{i} \frac{∂f}{∂u}) \frac{∂}{∂u_{i}} . \end{aligned}  (3.27)

Thus, starting with a field (3.27) in the case dim π = 1 or with an arbitrary field on J^{0}(π) for dim π > 1 and using (3.25), we can obtain efficient expressions for Lie fields.

Remark 3.3. Note that in the multi-dimensional case dim π > 1 we can introduce the functions f^{j} = i_{X}ω^{j}, where ω^{j} = du^{j} − ∑_{i} u_{i}^{j} dx_{i} are the Cartan forms on J^{1}(π). Such a function may be understood as an element of the module F_{1}(π, π). The local conditions of a section f ∈ F_{1}(π, π) to generate a Lie field is as follows:

\frac{∂f^{α}}{∂u_{i}^{α}} = \frac{∂f^{β}}{∂u_{i}^{β}} , \frac{∂f^{α}}{∂u_{i}^{β}} = 0, α ≠ β.

We call f the generating function (though, strictly speaking, the term generating section should be used) of the Lie field X, if X is a lifting of the field X_{f}.

Let us write down the conditions of a Lie field to be a symmetry. Assume that an equation E is given by the relations F^{1} = 0, …, F^{r} = 0, where F^{j} ∈ F_{k}(π). Then X is a symmetry of E if and only if

X(F^{j}) = ∑_{α=1}^{r} λ_{α}^{j}F^{α}, j = 1, …, r,

where λ_{α}^{j} are smooth functions, or X(F^{j}) |_{E} = 0, j = 1, …, r. These conditions can be rewritten in terms of generating sections and we shall do it later in a more general situation.

3.7. Prolongations of differential equations. Prolongations are differential consequences of a given differential equation. Let us give a formal definition.

Definition 3.19. Let E ⊂ J^{k}(π) be a differential equation of order k. Define the set

E^{1} = { θ_{k+1} ∈ J^{k+1}(π) | π_{k+1,k}(θ_{k+1}) ∈ E, L_{θ_{k+1}} ⊂ T_{π_{k+1,k}(θ_{k+1})}E }

and call it the first prolongation of the equation E.

If the first prolongation E^{1} is a submanifold in J^{k+1}(π), we define the second prolongation of E as (E^{1})^{1} ⊂ J^{k+2}(π), etc. Thus the l-th prolongation is a subset E^{l} ⊂ J^{k+l}(π).

Let us redefine the notion of l-th prolongation directly. Namely, take a point θ_{k} ∈ E and consider a section φ ∈ Γ_{loc}(π) such that the graph of j_{k}(φ) is tangent to E with order l. Let π_{k}(θ_{k}) = x ∈ M. Then [φ]_{x}^{k+l} is a point of J^{k+l}(π) and the set of all points obtained in such a way obviously coincides with E^{l}, provided all intermediate prolongations E^{1}, …, E^{l−1} be well defined in the sense of Definition 3.19.

Assume now that locally E is given by the equations F^{1} = 0, …, F^{r} = 0, F^{j} ∈ F_{k}(π) and θ_{k} ∈ E is the origin of the chosen special coordinate system. Let u^{1} = φ^{1}(x_{1}, …, x_{n}), …, u^{m} = φ^{m}(x_{1}, …, x_{n}) be a local section of the bundle π. Then the equations of the first prolongation are

\frac{∂F^{j}}{∂x_{i}} + ∑_{α,σ} \frac{∂F^{j}}{∂u_{σ}^{α}} u_{σi}^{α} = 0, i = 1, …, n, j = 1, …, r,

combined with the initial equations F^{r} = 0. From here and by comparison with the coordinate representation of prolongations for nonlinear differential operators (see Subsection 3.2), we obtain the following result:

Proposition 3.14. Let E ⊂ J^{k}(π) be a differential equation. Then

(1) If the equation E is determined by a differential operator Δ: Γ(π) → Γ(π′), then its l-th prolongation is given by the l-th prolongation Δ^{(l)} : Γ(π) → Γ(π′_{l}) of the operator Δ.

(2) If E is locally described by the system F^{1} = 0, …, F^{r} = 0, F^{j} ∈ F_{k}(π), then the system

D_{σ}F^{j} = 0, |σ| ≤ l, j = 1, …, r,  (3.28)

where D_{σ} = D_{i_{1}} ∘ ⋯ ∘ D_{i_{|σ|}}, σ = i_{1} … i_{|σ|}, corresponds to E^{l}. Here D_{i} stands for the i-th total derivative (see (3.26) on page 52).

From the definition it follows that for any l ≥ l′ ≥ 0 one has π_{k+l,k+l′}(E^{l}) ⊂ E^{l′} and consequently one has the maps π_{k+l,k+l′}: E^{l} → E^{l′}.

Definition 3.20. An equation E ⊂ J^{k}(π) is called formally integrable, if (1) all prolongations E^{l} are smooth manifolds and

(2) all the maps π_{k+l+1,k+l}: E^{l+1} → E^{l} are smooth fiber bundles.

Definition 3.21. The inverse limit proj lim_{l→∞} E^{l} with respect to projections π_{l+1,l} is called the infinite prolongation of the equation E and is denoted by E^{∞} ⊂ J^{∞}(π).

3.8. Basic structures on infinite prolongations. Let π: E → M be a vector bundle and E ⊂ J^{k}(π) be a k-th order differential equation. Then we have embeddings ε_{l}: E^{l} ⊂ J^{k+l}(π) for all l ≥ 0. Since, in general, the sets E^{l} may not be smooth manifolds, we define a function on E^{l} as the restriction f |_{E^{l}} of a smooth function f ∈ F_{k+l}(π). The set F_{l}(E) of all functions on E^{l} forms an ℝ-algebra in a natural way and ε_{l}^{∗}: F_{k+l}(π) → F_{l}(E) is a homomorphism of algebras. In the case of formally integrable equations, the algebra F_{l}(E) coincides with C^{∞}(E^{l}). Let I_{l} = ker ε_{l}^{∗}. Evidently, I_{l}(E) ⊂ I_{l+1}(E). Then I(E) = ⋃_{l≥0} I_{l}(E) is an ideal in F(π) which is called the ideal of the equation E. The function algebra on E^{∞} is the quotient algebra F(E) = F(π)/I(E) and coincides with inj lim_{l→∞} F_{l}(E) with respect to the system of homomorphisms π_{k+l+1,k+l}^{∗}. For all l ≥ 0, we have the homomorphisms ε_{l}^{∗}: F_{l}(E) → F(E). When E is formally integrable, they are monomorphic, but in any case the algebra F(E) is filtered by the images of ε_{l}^{∗}.

To construct differential calculus on E^{∞}, one needs the general algebraic scheme exposed in Section 1 and applied to the filtered algebra F(E). However, in the case of formally integrable equations, due to the fact that all E^{l} are smooth manifolds, this scheme may be simplified and combined with a purely geometrical approach (cf. with similar constructions of Subsection 3.3).

In special coordinates the infinite prolongation of the equation E is determined by the system similar to (3.28) with the only difference that |σ| is unlimited now. Thus, the ideal I(E) is generated by the functions D_{σ}F^{j}, |σ| ≥ 0, j = 1, …, m. From these remarks we obtain the following important fact.

Remark 3.4. Let E be a formally integrable equations. Then from the above said it follows that the ideal I(E) is stable with respect to the action of the total derivatives D_{i}, i = 1, …, n. Consequently, the restrictions D_{i}^{E} = D_{i} |_{E} : F(E) → F(E) are well defined and D_{i}^{E} are filtered derivations. In other words, we obtain that the vector fields D_{i} are tangent to any infinite prolongation and thus determine vector fields on E^{∞}. We shall often skip the superscript E in the notation of the above defined restrictions.

Example 3.9. Consider a system of evolution equations of the form

u_{t}^{j} = f^{j}(x, t, …, u^{α}, …, u_{x}^{α}, …), j, α = 1, …, m.

Then the set of functions x_{1}, …, x_{n}, t, …, u_{i_{1},...,i_{r},0}^{j} with 1 ≤ i_{k} ≤ n, j = 1, …, m, where t = x_{n+1}, may be taken for internal coordinates on E^{∞}. The total derivatives restricted to E^{∞} are expressed as

\begin{matrix}D_{i} = & \frac{∂}{∂x_{i}} + ∑_{j=1}^{n} ∑_{|σ|≥0} u_{σi}^{j} \frac{∂}{∂u_{σ}^{j}} , i = 1, …, n, \\ D_{t} = & \frac{∂}{∂t} + ∑_{j=1}^{n} ∑_{|σ|≥0} D_{σ}(f^{j})\frac{∂}{∂u_{σ}^{j}} \end{matrix}  (3.29)

in these coordinates, while the Cartan forms restricted to E^{∞} are written down as

ω_{σ}^{j} = du_{σ}^{j} − ∑_{i=1}^{n} u_{σi}^{j} dx_{i} − D_{σ}(f^{j}) dt.  (3.30)

Let π: E → M be a vector bundle and E ⊂ J^{k}(π) be a formally integrable equation.

Definition 3.22. Let θ ∈ J^{∞}(π). Then

(1) The Cartan plane C_{θ} = C_{θ}(π) ⊂ T_{θ}J^{∞}(π) at θ is the linear envelope of tangent planes to all manifolds j_{∞}(φ)(M), φ ∈ Γ(π), passing through θ.

(2) If θ ∈ E^{∞}, then the intersection C_{θ}(E) = C_{θ}(π)∩T_{θ}E^{∞} is called Cartan

plane of E^{∞} at θ. The correspondence θ ↦ C_{θ}(π), θ ∈ J^{∞}(π) (respectively, θ ↦ C_{θ}(E^{∞}), θ ∈ E^{∞}) is called the Cartan distribution on J^{∞}(π) (respectively, on E^{∞}).

Proposition 3.15. For any vector bundle π: E → M and a formally integrable equation E ⊂ J^{k}(π) one has:

(1) The Cartan plane C_{θ}(π) is n-dimensional at any point θ ∈ J^{∞}(π).

(2) Any point θ ∈ E^{∞} is generic, i.e., C_{θ}(π) ⊂ T_{θ}E^{∞} and thus one has C_{θ}(E^{∞}) = C_{θ}(π).

(3) Both distributions, C(π) and C(E^{∞}), are integrable.

Proof. Let θ ∈ J^{∞}(π) and π_{∞}(θ) = x ∈ M. Then the point θ completely determines all partial derivatives of any section φ ∈ Γ_{loc}(π) such that its graph passes through θ. Consequently, all such graphs have a common tangent plane at this point, which coincides with C_{θ}(π). This proves the first statement.

To prove the second one, recall Remark 3.4 on the preceding page: locally, any vector field D_{i} is tangent to E^{∞}. But as it follows from (3.20) on page 45, i_{D_{i}}ω_{σ}^{j} = 0 for any D_{i} and for any Cartan form ω_{σ}^{j}. Hence, linear independent vector fields D_{1}, …, D_{n} locally lie both in C(π) and in C(E^{∞}) which gives the result.

Finally, as it follows from the above said, the module

CD(π) = { X ∈ D(π) | X lies in C(π) }  (3.31)

is locally generated by the fields D_{1}, …, D_{n}. But it is easily seen that [D_{α}, D_{β}] = 0 for all α, β = 1, …, n and consequently [CD(π), CD(π)] ⊂ CD(π). The same reasoning is valid for

CD(E) = { X ∈ D(E^{∞}) | X lies in C(E^{∞}) }.  (3.32)

This completes the proof of the proposition.

Proposition 3.16. Maximal integral manifolds of the Cartan distribution C(π) are graph of infinite jets of sections j_{∞}(φ), φ ∈ Γ_{loc}(π).

Proof. Note first that graphs of infinite jets are integral manifolds of the Cartan distribution of maximal dimension (equaling to n) and that any integral manifold projects to J^{k}(π) and M without singularities.

Let now N ⊂ J^{∞}(π) be an integral manifold and N^{k} = π_{∞,k}N ⊂ J^{k}(π), N′ = π_{∞}N ⊂ M. Hence, there exists a diffeomorphism φ′: N′ → N^{0} such that π ∘ φ′ = id_{N′}. Then by the Whitney theorem on extension for smooth functions (see [38]), there exists a local section φ: M → E satisfying φ |_{N′} = φ′ and j_{k}(φ)(M) ⊃ N^{k} for all k > 0. Consequently, j_{∞}(φ)(M) ⊃ N.

Corollary 3.17. Maximal integral manifolds of the Cartan distribution on E^{∞} coincide locally with graphs of infinite jets of solutions.

Consider a point θ ∈ J^{∞}(π) and let x = π_{∞}(θ) ∈ M be its projection to M. Let v be a tangent vector to M at the point x. Then, since the Cartan plane C_{θ} isomorphically projects to T_{x}M, there exists a unique tangent vector Cv ∈ T_{θ}J^{∞}(π) such that (π_{∞})_{∗}(Cv) = v. Hence, for any vector field X ∈ D(M) we can define a vector field CX ∈ D(π) by setting (CX)_{θ} = C(X_{π_{∞}(θ)}). Then, by construction, the field CX is projected by (π_{∞})_{∗} to X while the correspondence C: D(M) → D(π) is a C^{∞}(M)-linear one. In other words, this correspondence is a linear connection in the bundle π_{∞}: J^{∞}(π) → M.

Definition 3.23. The connection C: D(M) → D(π) defined above is called the Cartan connection in J^{∞}(π).

For any formally integrable equation, the Cartan connection is obviously restricted to the bundle π_{∞}: E^{∞} → M and we preserve the same notation C for this restriction.

Let (x_{1}, …, x_{n}, …, u_{σ}^{j}, …) be a special coordinate system in J^{∞}(π) and X = X_{1}∂/∂x_{1} + ⋯ + X_{n}∂/∂x_{n} be a vector field on M represented in this coordinate system. Then the field CX is to be of the form CX = X + X^{v}, where X^{v} = ∑_{j,σ} X_{σ}^{j}∂/∂u_{σ}^{j} is a π_{∞}-vertical field. The defining conditions i_{CX}ω_{σ}^{j} = 0, where ω_{σ}^{j} are the Cartan forms on J^{∞}(π), imply

CX = ∑_{i=1}^{n} X_{i} (\frac{∂}{∂x_{i}} + ∑_{j,σ} u_{σi}^{j} \frac{∂}{∂u_{σ}^{j}} ) = ∑_{i=1}^{n} X_{i}D_{i}.  (3.33)

In particular, C(∂/∂x_{i}) = D_{i}, i.e., total derivatives are the liftings to J^{∞}(π) of the corresponding partial derivatives by the Cartan connection.

Let now V be a vector field on E^{∞} and θ ∈ E^{∞} be a point. Then the vector V_{θ} can be projected parallel to the Cartan plane C_{θ} to the fiber of the projection π_{∞}: E^{∞} → M passing through θ. Thus we get a vertical vector field V^{v}. Hence, for any f ∈ F(E) a differential one-form U_{C}(f) ∈ Λ^{1}(E) is defined by

i_{V} (U_{C}(f)) = V^{v}(f), V ∈ D(E).  (3.34)

The correspondence f ↦ U_{C}(f) is a derivation of the algebra F(E) with the values in the F(E)-module Λ^{1}(E), i.e., U_{C}(fg) = fU_{C}(g) + gU_{C}(f) for all f, g ∈ F(E).

Definition 3.24. The derivation U_{C}: F(E) → Λ^{1}(E) is called the structural element of the equation E.

In the case E^{∞} = J^{∞}(π) the structural element U_{C} is locally represented in the form

U_{C} = ∑_{j,σ} ω_{σ}^{j} ⊗ \frac{∂}{∂u_{σ}^{j}} ,  (3.35)

where ω_{σ}^{j} are the Cartan forms on J^{∞}(π). To obtain the expression in the general case, one needs to rewrite (3.35) in local coordinates.

The following result is a consequence of definitions:

Proposition 3.18. For any vector field X ∈ D(M) the equality

j_{∞}(φ)^{∗}(CX(f)) = X(j_{∞}(φ)^{∗}(f)), f ∈ F(π), φ ∈ Γ_{loc}(π), (3.36)

takes place. Equality (3.36) uniquely determines the Cartan connection in J^{∞}(π).

Corollary 3.19. The Cartan connection in E^{∞} is flat, i.e., C[X, Y] = [CX, CY] for any X, Y ∈ D(M).

Proof. Consider the case E^{∞} = J^{∞}(π). Then from Proposition 3.18 on the preceding page we have

j_{∞}(φ)^{∗}(C[X, Y](f)) = [X, Y](j_{∞}(φ)^{∗}(f))

= X(Y (j_{∞}(φ)^{∗}(f))) − Y (X(j_{∞}(φ)^{∗}(f)))

for any φ ∈ Γ_{loc}(π), f ∈ F(π). On the other hand,

j_{∞}(φ)^{∗}([CX, CY](f)) = j_{∞}(φ)^{∗}(CX(CY (f)) − CY (CX(f))) = X(j_{∞}(φ)^{∗}(Y (f))) − Y (j_{∞}(φ)^{∗}(CX(f)))

= X(Y (j_{∞}(φ)^{∗}(f))) − Y (X(j_{∞}(φ)^{∗}(f))).

To prove the statement for an arbitrary formally integrable equation E, it suffices to note that the Cartan connection in E^{∞} is obtained by restricting the fields CX to infinite prolongation of E.

The construction of Proposition 3.18 on the page before can be generalized. Let π: E → M be a vector bundle and ξ_{1} : E_{1} → M, ξ_{2} : E_{2} → M be another two vector bundles.

Definition 3.25. Let Δ: Γ(ξ_{1}) → Γ(ξ_{2}) be a linear differential operator. The lifting CΔ: F(π, ξ_{1}) → F(π, ξ_{2}) of the operator Δ is defined by

j_{∞}(φ)^{∗}(CΔ(f)) = Δ(j_{∞}(φ)^{∗}(f)),  (3.37)

where φ ∈ Γ_{loc}(π), f ∈ F(π, ξ_{1}) are arbitrary sections.

Proposition 3.20. Let π: E → M, ξ_{i}: E_{i} → M, i = 1, 2, 3, be vector bundles. Then

(1) For any C^{∞}(M)-linear differential operator Δ: Γ(ξ_{1}) → Γ(ξ_{2}), the operator CΔ is an F(π)-linear differential operator of the same order.

(2) For any Δ, □: Γ(ξ_{1}) → Γ(ξ_{2}) and f, g ∈ F(π), one has

C(fΔ + g□) = fCΔ + gC□.

(3) For Δ_{1} : Γ(ξ_{1}) → Γ(ξ_{2}) and Δ_{2} : Γ(ξ_{2}) → Γ(ξ_{3}), one has

C(Δ_{2} ∘ Δ_{1}) = CΔ_{2} ∘ CΔ_{1}.

From this proposition and from Proposition 3.18 on the preceding page it follows that if Δ is a scalar differential operator C^{∞}(M) → C^{∞}(M) locally represented as Δ = ∑_{σ} a_{σ}∂^{|σ|}/∂x_{σ}, a_{σ} ∈ C^{∞}(M), then CΔ = ∑_{σ} a_{σ}D_{σ} in the corresponding special coordinates. If Δ = ∥Δ_{ij}∥ is a matrix operator, then CΔ = ∥CΔ_{ij}∥. Obviously, CΔ may be understood as a constant differential operator acting from sections of the bundle π to linear differential operators from Γ(ξ_{1}) to Γ(ξ_{2}). This observation is generalized as follows.

Definition 3.26. An F(π)-linear differential operator Δ acting from the module F(π, ξ_{1}) to F(π, ξ_{2}) is called a C-differential operator, if it admits restriction to graphs of infinite jets, i.e., if for any section φ ∈ Γ(π) there exists an operator Δ_{φ}: Γ(ξ_{1}) → Γ(ξ_{2}) such that

j_{∞}(φ)^{∗}(Δ(f)) = Δ_{φ}(j_{∞}(φ)^{∗}(f))  (3.38)

for all f ∈ F(π, ξ_{1}).

Thus, C-differential operators are nonlinear differential operators taking their values in C^{∞}(M)-modules of linear differential operators.

Exercise 3.2. Consider a C-differential operator Δ: F(π, ξ_{1}) → F(π, ξ_{2}). Prove that if Δ(π^{∗}(f)) = 0 for all f ∈ Γ(ξ_{1}), then Δ = 0.

Proposition 3.21. Let π, ξ_{1}, and ξ_{2} be vector bundles over M. Then any C-differential operator Δ: F(π, ξ_{1}) → F(π, ξ_{2}) can be presented in the form Δ = ∑_{α} a_{α}CΔ_{α}, a_{α} ∈ F(π), where Δ_{α} are linear differential operators acting from Γ(ξ_{1}) to Γ(ξ_{2}).

Proof. Recall that we consider the filtered theory; in particular, there exists an integer l such that Δ(F_{k}(π, ξ_{1})) ⊂ F_{k+l}(π, ξ_{2}) for all k. Consequently, since Γ(ξ_{1}) is embedded into F_{0}(π, ξ_{1}), we have Δ(Γ(ξ_{1})) ⊂ F_{l}(π, ξ_{2}) and the restriction Δ̄ = Δ |_{Γ(ξ_{1})} is a C^{∞}(M)-differential operator taking its values in F_{l}(π, ξ_{2}).

On the other hand, the operator Δ̄ is represented in the form Δ̄ = ∑_{α} a_{α}Δ_{α}, a_{α} ∈ F_{l}(π), with Δ_{α}: Γ(ξ_{1}) → Γ(ξ_{2}) being C^{∞}(M)-linear differential operators. Define CΔ̄ = ∑_{α} a_{α}CΔ_{α}. Then the difference Δ − CΔ̄ is a C-differential operator such that its restriction to Γ(ξ_{1}) vanishes. Therefore, by Exercise 3.2 Δ = CΔ̄.

Corollary 3.22. C-differential operators admit restrictions to infinite prolongations: if Δ: F(π, ξ_{1}) → F(π, ξ_{2}) is a C-differential operator and E ⊂ J^{k}(π) is a k-th order equation, then there exists a linear differential operator Δ_{E}: F(E, ξ_{1}) → F(E, ξ_{2}) such that ε^{∗} ∘ Δ = Δ_{E} ∘ ε^{∗}, where ε: E^{∞} ↪ J^{∞}(π) is the natural embedding.

Proof. The result immediately follows from Remark 3.4 on page 55 and from Proposition 3.21.

Example 3.10. Let ξ_{1} = τ_{i}^{∗}, ξ_{2} = τ_{i+1}^{∗}, where τ_{p}^{∗}: ⋀^{p} T^{∗}M → M (see Example 3.2 on page 38), and Δ = d: Λ^{i}(M) → Λ^{i+1}(M) be the de Rham differential. Then we obtain the first-order operator d̄ = Cd: Λ̄^{i}(π) → Λ̄^{i+1}(π), where Λ̄^{p}(π) denotes the module F(π, τ_{p}^{∗}). Due Corollary 3.22 the operators d̄: Λ̄^{i}(E) → Λ̄^{i+1}(E) are also defined, where Λ̄^{p}(E) = F(E, τ_{p}^{∗}).

Definition 3.27. Let E ⊂ J^{k}(π) be an equation.

(1) Elements of the module Λ̄^{i}(E) are called horizontal i-forms on the infinite prolongation E^{∞}.

(2) The operator d̄: Λ̄^{i}(E) → Λ̄^{i+1}(E) is called the horizontal de Rham differential on E^{∞}.

From Proposition 3.20 (3) on page 58 it follows that d̄∘d̄= 0. The sequence

0 ⟶ F(E) ⟶^{d̄} Λ̄^{1}(E) ⟶ ⋯ ⟶ Λ̄^{i}(E) ⟶^{d̄} Λ̄^{i+1}(E) → ⋯

is called the horizontal de Rham complex of the equation E. Its cohomology is called the horizontal de Rham cohomology of E and is denoted by H̄^{∗}(E) = ⨁_{i≥0} H̄^{i}(E).

In local coordinates, horizontal forms of degree p on E^{∞} are represented as ω = ∑_{i_{1}<···<i_{p}} a_{i_{1}...i_{p}}dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{p}}, where a_{i_{1}...i_{p}} ∈ F(E), while the horizontal de Rham differential acts as

d̄(ω) = ∑_{i=1}^{n} ∑_{i_{1}<···<i_{p}} D_{i}(a_{i_{1}...i_{p}}) dx_{i} ∧ dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{p}}.  (3.39)

In particular, we see that both Λ̄^{i}(E) and H̄^{i}(E) vanish for i > dim M.

Consider the algebra Λ^{∗}(E) of all differential forms on E^{∞} and note that one has the embedding Λ̄^{∗}(E) ↪ Λ^{∗}(E). Let us extend the horizontal de Rham differential to this algebra as follows:

d̄(dω) = −d(d̄(ω)), d̄(ω ∧ θ) = d̄(ω) ∧ θ + (−1)^{p}ω ∧ d̄(θ) ω ∈ Λ^{p}(E).

Obviously, these conditions define the differential d̄: Λ^{i}(E) → Λ^{i+1}(E) and its restriction to Λ̄^{∗}(E) coincides with the horizontal de Rham differential.

Let us also set d_{C} = d − d̄: Λ^{∗}(E) → Λ^{∗}(E) and call d_{C} the Cartan (or vertical) differential on E^{∞}. Then from definitions we obtain

d = d̄+ d_{C}, d̄∘ d̄= d_{C} ∘ d_{C} = 0, d_{C} ∘ d̄+ d̄∘ d_{C} = 0,

i.e., the pair (d̄, d_{C}) forms a bicomplex in Λ^{∗}(E) with the total differential d. It is called the variational bicomplex and will be discussed in more details in Section 7.

Denote by CΛ^{1}(E) the Cartan submodule in Λ^{1}(E), i.e., the module of 1forms vanishing on the Cartan distribution on E^{∞} (cf. with Definition 3.12 on page 46). Then the splitting d = d̄+ d_{C} implies the direct sum decomposition

Λ^{1}(E) = Λ̄^{1}(E) ⊕ CΛ^{1}(E),

which gives

Λ^{i}(E) = ⨁_{p+q=i} Λ̄^{q}(E) ⊗_{F(E)} C^{p}Λ(E),  (3.40)

where C^{p}Λ(E) = \underbrace{CΛ^{1}(E) ∧ ⋯ ∧ CΛ^{1}(E)}_{p times}.

To conclude this section, we shall write down the coordinate representation for the Cartan differential d_{C} and the extended differential d̄. First note that by definition and due to representation (3.39) on the facing page, one has

d_{C}(f) = ∑_{j,σ} \frac{∂f}{∂u_{σ}^{j}} ω_{σ}^{j}, f ∈ F(π).  (3.41)

In particular, d_{C} takes coordinate functions u_{σ}^{j} to the corresponding Cartan forms. This is reason why we called d_{C} the Cartan differential on E^{∞}. It is easily seen that d_{C} |_{F(E)} = U_{C}(E) (see Definition 3.24 on page 57). To finish computations, it suffices to compute d̄(ω_{σ}^{j}):

d̄(ω_{σ}^{j}) = d̄d_{C}(u_{σ}^{j}) = −d_{C}d̄(u_{σ}^{j})

and thus

d̄(ω_{σ}^{j}) = − ∑_{i=1}^{n} ω_{σi}^{j} ∧ dx_{i}.  (3.42)

Note that from the results obtained it follows that

d̄(Λ̄^{q}(E) ⊗ C^{p}Λ(E)) ⊂ Λ̄^{q+1}(E) ⊗ C^{p}Λ(E),
d_{C}(Λ̄^{q}(E) ⊗ C^{p}Λ(E)) ⊂ Λ̄^{q}(E) ⊗ C^{p+1}Λ(E).

Now let us define the module of horizontal jets. Let ξ be a vector bundle over M. Say that two elements of P = F(E, ξ) are horizontally equivalent up to order k ≤ ∞ at point θ ∈ E^{∞}, if their total derivatives up to order k coincide at θ. The horizontal jet space J̄_{θ}^{k}(P) is P modulo this relation, and the collection J̄^{k}(P) = ⋃_{θ ∈ E^{∞}} J̄_{θ}^{k}(P) constitutes the horizontal jet bundle J̄^{k}(P) → E^{∞}. We denote the module of sections of horizontal jet bundle by J̄^{k}(P).

As with the usual jet bundles, there exist the natural C-differential operators

j̄_{k}: P → J̄^{k}(P),

and the natural projections ν_{k,l}: J̄^{k}(P) → J̄^{l}(P) such that ν_{k,l} ∘ j̄_{k} = j̄_{l}. The operators j̄_{k} and ν_{k,l} are restrictions of the operators Cj_{k} and Cπ_{k,l}^{∗} to E^{∞}.

C-differential operators, horizontal forms and jets constitute a “subtheory” in the differential calculus on an infinitely prolonged equation. It is, roughly speaking, “the total derivatives calculus” and is called C-differen-tial calculus. It is easily shown that all components of usual calculus and the Lagrangian formalism discussed above have their counterparts in the framework of C-differential calculus. All constructions of Sections 1 and 2 are carried over into C-differential calculus word for word as long as the operators, jets, and forms in them are assumed respectively C-differential and horizontal.

3.9. Higher symmetries. Let π: E → M be a vector bundle and E ⊂ J^{k}(π) be a differential equation. We shall still assume E to be formally integrable, though it not restrictive in this context.

Consider a symmetry F : J^{k}(π) ⟶ J^{k}(π) of the equation E and let θ_{k+1} be a point of the first prolongation E^{1} such that π_{k+1,k}(θ_{k+1}) = θ_{k} ∈ E. Then the R-plane L_{θ_{k+1}} is taken to the R-plane F_{∗}(L_{θ_{k+1}}), since F is a Lie transformation, and F_{∗}(L_{θ_{k+1}}) ⊂ T_{F (θ_{k})}, since F is a symmetry. Consequently, the lifting F^{(1)} : J^{k+1}(π) → J^{k+1}(π) is a symmetry of E^{1}. By the same reasons, F^{(l)} is a symmetry of the l-th prolongation of E. From here it also follows that for any infinitesimal symmetry X of the equation E, its l-th lifting is is a symmetry of E^{l} as well.

Proposition 3.23. Symmetries of a formally integrable equation E ⊂ J^{k}(π) coincide with symmetries of any prolongation of this equation. The same is valid for infinitesimal symmetries.

Proof. We showed already that to any (infinitesimal) symmetry of E there corresponds an (infinitesimal) symmetry of E^{l}. Consider an (infinitesimal) symmetry of E^{l}. By Theorems 3.12 on page 50 and 3.13 on page 51, it is π_{k+l,k}-fiberwise and therefore generates an (infinitesimal) symmetry of the equation E.

The result proved means that a symmetry of E generates a symmetry of E^{∞} which preserves every prolongation of finite order. A natural step to generalize the concept of symmetry is to consider “all symmetries” of E^{∞}. Recall the notation

CD(π) = { X ∈ D(π) | X lies in C(π) }

(cf. with (3.23) on page 51).

Definition 3.28. Let π be a vector bundle. A vector field X ∈ D(π) is called a symmetry of the Cartan distribution C(π) on J^{∞}(π), if [X, CD(π)] ⊂ CD(π).

Thus, the set of symmetries coincides with D_{C}(π) (see (3.24) on page 51) and forms a Lie algebra over ℝ and a module over F(π). Note that since the Cartan distribution on J^{∞}(π) is integrable, one has CD(π) ⊂ D_{C}(π) and, moreover, CD(π) is an ideal in the Lie algebra D_{C}(π).

Note also that symmetries belonging to CD(π) are tangent to any integral manifold of the Cartan distribution. By this reason, we call such symmetries trivial. Respectively, the elements of the quotient Lie algebra

sym(π) = D_{C}(π)/CD(π)

are called nontrivial symmetries of the Cartan distribution on J^{∞}(π).

Let now E^{∞} be the infinite prolongation of an equation E ⊂ J^{k}(π). Then, since CD(π) is spanned by the fields of the form CY, where Y ∈ D(M) (see Remark 3.4 on page 55), any vector field from CD(π) is tangent to E^{∞}. Consequently, either all elements of the coset [X] = X mod CD(π), X ∈ D(π), are tangent to E^{∞} or neither of them do. In the first case we say that the coset [X] is tangent to E^{∞}.

Definition 3.29. An element [X] = X mod CD(π), X ∈ D(π), is called a higher symmetry of E, if it is tangent to E^{∞}.

The set of all higher symmetries forms a Lie algebra over ℝ and is denoted by sym(E). We shall usually omit the adjective higher in the sequel.

Consider a vector field X ∈ D(π). Then, substituting X into the structural element U_{C} (see (3.35) on page 57), we obtain a field X^{v} ∈ D(π). The correspondence U_{C}: X ↦ X^{v} = i_{X}U_{C} possesses the following properties:

(1) The field X^{v} is vertical, i.e., X^{v}(C^{∞}(M)) = 0.

(2) X^{v} = X for any vertical field.

(3) X^{v} = 0 if and only if the field X lies in CD(π). Therefore, we obtain the direct sum decomposition of F(π)-modules

D(π) = D^{v}(π) ⊕ CD(π),

where D^{v}(π) denotes the Lie algebra of vertical fields. A direct corollary of these properties is the following result.

Proposition 3.24. For any coset [X] ∈ sym(E) there exists a unique vertical representative and thus

sym(E) = { X ∈ D^{v}(E) | [X, CD(E)] ⊂ CD(E) },  (3.43)

where CD(E) is spanned by the fields CY, Y ∈ D(M).

Lemma 3.25. Let X ∈ sym(π) be a vertical vector field. Then it is completely determined by its restriction to F_{0}(π) ⊂ F(π).

Proof. Let X satisfy the conditions of the lemma and Y ∈ D(M). Then for any f ∈ C^{∞}(M) one has

[X, CY](f) = X(CY (f)) − CY (X(f)) = X(Y (f)) = 0

and hence the commutator [X, CY] is a vertical vector field. On the other hand, [X, CY] ∈ CD(π) because CD(π) is a Lie algebra ideal. Consequently, [X, CY] = 0. Note now that in special coordinates we have D_{i}(u_{σ}^{j}) = u_{σi}^{j} for all σ and j. From the above said it follows that

X(u_{σi}^{j}) = D_{i}(X(u_{σ}^{j})).  (3.44)

But a vertical derivation is determined by its values at the coordinate functions u_{σ}^{j}.

Let now X_{0} : F_{0}(π) → F(π) be a derivation. Then equalities (3.44) on the preceding page allow one to reconstruct locally a vertical derivation X ∈ D(π) satisfying X |_{F_{0}(π)} = X_{0}. Obviously, the derivation X lies in sym(π) over the neighborhood under consideration. Consider two neighborhoods U_{1}, U_{2} ⊂ J^{∞}(π) with the corresponding special coordinates in each of them and two symmetries X^{i} ∈ sym(π |_{U_{i}}), i = 1, 2, arising by the described procedure. But the restrictions of X^{1} and X^{2} to F_{0}(π |_{U_{1}∩U_{2}} ) coincide. Hence, by Lemma 3.25 on the page before, the field X^{1} coincides with X^{2} over the intersection U_{1} ∩ U_{2}. Hence, the reconstruction procedure X_{0} ↦ X is a global one. So we have established a one-to-one correspondence between elements of sym(π) and derivations F_{0}(π) → F(π).

Note now that due to vector bundle structure in π: E → M, derivations F_{0}(π) → F(π) are identified with sections of π_{∞}^{∗}(π), or with elements of F(π, π).

Theorem 3.26. Let π: E → M be a vector bundle. Then:

(1) The F(π)-module sym(π) is in one-to-one correspondence with elements of the module F(π, π).

(2) In special coordinates the correspondence F(π, π) → sym(π) is expressed by the formula^{7}

φ ↦ �_{φ} = ∑_{j,σ} D_{σ}(φ^{j})\frac{∂}{∂u_{σ}^{j}} ,  (3.45)

where φ = (φ^{1}, …, φ^{m}) is the component-wise representation of the section φ ∈ F(π, π).

Proof. The first part of the theorem has already been proved. To prove the second one, it suffices to use equality (3.44) on the page before.

Definition 3.30. Let π: E → M be a vector bundle.

(1) The field �_{φ} of the form (3.45) is called an evolutionary vector field on J^{∞}(π).

(2) The section φ ∈ F(π, π) is called the generating function of the field �_{φ}.

Remark 3.5. Let ζ: N → M be an arbitrary smooth fiber bundle and ξ: P → M be a vector bundle. Then it easy to show that any ζvertical vector field X on N can be uniquely lifted up to an ℝ-linear map X^{ξ}: Γ(ζ^{∗}(ξ)) → Γ(ζ^{∗}(ξ)) such that

X^{ξ}(fψ) = X(f)ψ + fX^{ξ}(ψ), f ∈ C^{∞}(N), ψ ∈ Γ(ζ^{∗}(ξ)). (3.46)

In particular, taking π_{∞} for ζ, for any evolutionary vector field �_{φ} we obtain the family of maps �_{φ}^{ξ}: F(π, ξ) → F(π, ξ) satisfying (3.46).

7To denote evolutionary vector fields (see Definition 3.30), we use the Cyrillic letter �, which is pronounced like “e” in “ten”.

Consider the map �_{φ}^{π}: F(π, π) → F(π, π) and recall the element ρ_{0} ∈ F_{0}(π, π) ⊂ F(π, π) (see Example 3.1 on page 38). It can be easily seen that

�_{φ}^{π}(ρ_{0}) = φ  (3.47)

which can be taken for the definition of the generating section.

Let �_{φ}, �_{ψ} be two evolutionary vector fields. Then, since sym(π) is a Lie algebra and by Theorem 3.26 on the preceding page, there exists a unique section {φ, ψ} satisfying [�_{φ}, �_{ψ}] = �_{{φ,ψ}}.

Definition 3.31. The section {φ, ψ} ∈ F(π, π) is called the (higher) Jacobi bracket of the sections φ, ψ ∈ F(π).

Proposition 3.27. Let φ, ψ ∈ F(π, π) be two sections. Then:

(1) {φ, ψ} = �_{φ}^{π}(ψ) − �_{ψ}^{π}(φ).

(2) In special coordinates, the Jacobi bracket of φ and ψ is expressed by the formula

\{φ, ψ\}^{j} = ∑_{α,σ} (D_{σ}(φ^{α})\frac{∂ψ^{j}}{∂u_{σ}^{α}} − D_{σ}(ψ^{α})\frac{∂φ^{j}}{∂u_{σ}^{α}} ) ,  (3.48)

where superscript j denotes the j-th component of the corresponding section.

Proof. To prove (1), let us use (3.47):

{φ, ψ} = �_{{φ,ψ}}^{π}(ρ_{0}) = �_{φ}^{π}(�_{ψ}^{π}(ρ_{0})) − �_{ψ}^{π}(�_{φ}^{π}(ρ_{0})) = �_{φ}^{π}(ψ) − �_{ψ}^{π}(φ).

The second statement follows from the first one and from equality (3.45) on the preceding page.

Consider now a nonlinear operator Δ: Γ(π) → Γ(ξ) and let ψ_{Δ} ∈ F(π, ξ) be the corresponding section. Then for any φ ∈ F(π, π) the section �_{φ}^{ξ}(ψ_{Δ}) ∈ F(π, ξ) is defined and we can set

ℓ_{Δ}(φ) = �_{φ}^{ξ}(ψ_{Δ}).  (3.49)

Definition 3.32. The operator ℓ_{Δ} : F(π, π) → F(π, ξ) defined by (3.49) is called the universal linearization operator^{8} of the operator Δ: Γ(π) → Γ(ξ).

From the definition and equality (3.45) on the preceding page we obtain that for a scalar differential operator

Δ: φ ↦ F (x_{1}, …, x_{n}, …, \frac{∂^{|σ|}φ^{j}}{∂x_{σ}} , …)

one has ℓ_{Δ} = (ℓ_{Δ}^{1}, …, ℓ_{Δ}^{m}), m = dim π, where

ℓ_{Δ}^{α} = ∑_{σ} \frac{∂F}{∂u_{σ}^{α}} D_{σ}.  (3.50)

8Cf. with the algebraic definition on page 34.

If dim ξ = r > 1 and Δ = (Δ_{1}, …, Δ_{r}), then

ℓ_{Δ} = ∥ℓ_{Δ^{α}}^{β}∥ , α = 1, …, m, β = 1, …, r.  (3.51)

In particular, we see that the following statement is valid.

Proposition 3.28. For any differential operator Δ, its universal linearization is a C-differential operator.

Now we can describe the algebra sym(E), E ⊂ J^{k}(π) being a formally integrable equation. Let I(E) ⊂ F(π) be the ideal of the equation E (see page 54). Then, by definition, �_{φ} is a symmetry of E if and only if

�_{φ}(I(E)) ⊂ I(E).  (3.52)

Assume now that E is given by a differential operator Δ: Γ(π) → Γ(ξ) and locally is described by the system of equations F^{1} = 0, …, F^{r} = 0, F^{j} ∈ F(π). Then the functions F^{1}, …, F^{r} are differential generators of the ideal I(E) and condition (3.52) may be rewritten as

�_{φ}(F^{j}) = ∑_{α,σ} a_{σ,j}^{α}D_{σ}(F^{α}), j = 1, …, m, a_{σ}^{α} ∈ F(π).  (3.53)

Using of (3.49) on the page before, the last equation acquires the form^{9}

ℓ_{F^{j}}(φ) = ∑_{α,σ} a_{σ,j}^{α}D_{σ}(F^{α}), j = 1, …, m, a_{σ}^{α} ∈ F(π).  (3.54)

But by Proposition 3.28, the universal linearization is a C-differential operator and consequently can be restricted to E^{∞} (see Corollary 3.22 on page 59). It means that we can rewrite (3.54) as

ℓ_{F^{j}} |_{E^{∞}} (φ |_{E^{∞}}) = 0, j = 1, …, m.  (3.55)

Combining these equations with (3.50) and (3.51), we obtain the following fundamental result:

Theorem 3.29. Let E ⊂ J^{k}(π) be a formally integrable equation and Δ = Δ_{E}: Γ(π) → Γ(ξ) be the operator corresponding to E. Then an evolutionary vector field �_{φ}, φ ∈ F(π, π) is a symmetry of E if and only if

ℓ_{E}(φ̄) = 0,  (3.56)

where ℓ_{E} and φ̄ denote restrictions of ℓ_{Δ} and φ on E^{∞} respectively. In other words, sym(E) = ker ℓ_{E}.

Exercise 3.3. Show that classical symmetries (see Subsection 3.6) are embedded in sym E as a Lie subalgebra. Describe their generating functions.

9We use the notation ℓ_{F}, F ∈ F(π, ξ), as a synonym for ℓ_{Δ}, where Δ: Γ(π) → Γ(ξ) is the operator corresponding to the section F.

Remark 3.6. From the result obtained it follows that higher symmetries of E can be identified with elements of F(E, π) satisfying equation (3.56) on the facing page. Below we shall say that a section φ ∈ F(E, π) is a symmetry of E keeping in mind this identification. Note that due to the fact that sym(E) is a Lie algebra, for any two symmetries φ, ψ ∈ F(E, π) their Jacobi bracket {φ, ψ}_{E} ∈ F(E, π) is well defined and is a symmetry as well. If no confusion arises, we shall omit the subscript E in the notation of the Jacobi bracket.

Finally, we give a useful description of the modules D^{v}(E) and C^{k}Λ(E). Denote κ = F(E, π).

First consider the case E^{∞} = J^{∞}(π). From the coordinate expression (3.45) on page 64 for an evolutionary vector field it immediately follows that any vertical tangent vector at point θ ∈ J^{∞}(π) can be realized in the form �_{φ}| _{θ} for some φ. This shows that the map φ ↦ �_{φ} yields the canonical isomorphism

D^{v}(π) = J̄^{∞}(κ).

The dual isomorphism reads

C^{1}Λ(π) = CDiff(κ, F).

In coordinates, this isomorphism takes the form ω_{σ}^{j} to the operator

(0, …, 0, D_{σ}, 0, …, 0),

with D_{σ} on j-th place.

It is clear that the Cartan k-forms can be identified with multilinear skew-symmetric C-differential operators in k arguments:

C^{p}Λ(π) = CDiff_{(p)}^{alt}(κ, F).

Now consider the general case. Suppose that the equation E is given by an operator Δ: Γ(π) → Γ(ξ). Denote P = F(E, ξ), so that ℓ_{E}: κ → P. From (3.55) on the facing page we get

Proposition 3.30. (1) The module D^{v}(E) is isomorphic to the kernel of

the homomorphism ψ_{∞}^{ℓ_{E}}: J̄^{∞}(κ) ⟶ J̄^{∞}(P);

(2) the module C^{p}Λ(E) is isomorphic to CDiff_{(p)}^{alt}(κ, F) modulo the submodule consisting of the operators of the form ∇ ∘ ℓ_{E}, where ∇ ∈ CDiff(P, CDiff_{(p−1)}^{alt}(κ, F)).

4. Coverings and nonlocal symmetries

The facts exposed in this section constitute a formal base to introduce nonlocal variables to the differential setting, i.e., variables of the type ∫ φ dx, φ being a function on an infinitely prolonged equation. A detailed exposition of this material can be found in [33] and [34].

4.1. Coverings. We start with fixing up the setting. To do it, extend the universum of infinitely prolonged equations in the following way. Let N be a chain of smooth maps ⋯ ⟶ N^{i+1} ⟶^{τ_{i+1,i}} N^{i} ⟶ ⋯, where N^{i} are smooth finite-dimensional manifolds. Define the algebra F(N) of smooth functions on N as the direct limit of the homomorphisms

⋯ ⟶ C^{∞}(N^{i}) ⟶^{τ_{i+1,i}^{∗}} C^{∞}(N^{i+1}) ⟶ · · ·. Then there exist natural homomorphisms τ_{∞,i}^{∗}: C^{∞}(N^{i}) → F(N) and the algebra F(N) may be considered to be filtered by the images of these maps. Let us consider calculus (see Section 1) over F(N) agreed with this filtration. Define the category Inf as follows:

(1) The objects of Inf are the above introduced chains N endowed with integrable distributions D_{N} ⊂ D(F(N)), where the word “integrable” means that [D_{N} , D_{N}] ⊂ D_{N}.

(2) If N_{1} = {N_{1}^{i}, τ_{i+1,i}^{1}}, N_{2} = {N_{2}^{i}, τ_{i+1,i}^{2}} are two objects of Inf, then a morphism φ: N_{1} → N_{2} is a system of smooth maps φ_{i}: N_{1}^{i+α} → N_{2}^{i}, where α ∈ ℤ is independent of i, satisfying τ_{i+1,i}^{2}∘φ_{i+1} = φ_{i}∘τ_{i+α+1,i+α}^{1} and such that φ_{∗,θ}(D_{N_{1},θ}) ⊂ D_{N_{2},φ(θ)} for any point θ ∈ N_{1}.

Definition 4.1. A morphism φ: N_{1} → N_{2} is called a covering in the category Inf, if φ_{∗,θ} |_{D_{N_{1},θ}} : D_{N_{1},θ} → D_{N_{2},φ(θ)} is an isomorphism for any point θ ∈ N_{1}.

In particular, manifolds J^{∞}(π) and E^{∞} endowed with the corresponding Cartan distributions are objects of Inf and we can consider coverings over these objects.

Example 4.1. Let Δ: Γ(π) → Γ(π′) be a differential operator of order ≤ k. Then the system of maps Φ_{Δ}^{(l)} : J^{l+l}(π) → J^{l}(π′) (see the proof of Lemma 3.3 on page 38) is a morphism of J^{∞}(π) to J^{∞}(π′). Under unrestrictive conditions of regularity, its image is of the form E^{∞} for some equation E in the bundle π′ while the map J^{∞}(π) → E^{∞} is a covering.

Definition 4.2. Let φ′: N′ → N and φ′′: N′′ → N be two coverings.

(1) A morphism ψ: N′ → N′′ is said to be a morphism of coverings, if φ′ = φ′′ ∘ ψ.

(2) The coverings φ′, φ′′ are called equivalent, if there exists a morphism ψ: N′ → N′′ which is a diffeomorphism.

Definition 4.3. A covering φ : N′ ⟶ N is called linear, if (1) φ is a linear bundle;

(2) any automorphism of the bundle φ is a morphism of the covering φ to itself.

Let N be an object of Inf and W be a smooth manifold. Consider the projection τ_{W} : N × W → N to the first factor. Then we can make a covering of τ_{W} by lifting the distribution D_{N} to N × W in a trivial way.

Definition 4.4. A covering τ: N′ → N is called trivial, if it is equivalent to the covering τ_{W} for some W.

Let again φ′: N′ → N, φ′′: N′′ → N be two coverings. Consider the commutative diagram

N′ ×_{N} N′′ ⟶^{φ′′^{∗}(φ′)} N′′
φ′^{∗}(φ′′)↓  ↓^{φ′′}
N′  ⟶^{φ′} N

where

N′ ×_{N} N′′ = { (θ′, θ′′) ∈ N′ × N′′ | φ′(θ′) = φ′′(θ′′) }

while φ′^{∗}(φ′′), φ′′^{∗}(φ′) are natural projections. The manifold N′ ×_{N} N′′ is supplied with a natural structure of an object of Inf and the maps (φ′)^{∗}(φ′′), (φ′′)^{∗}(φ′) become coverings.

Definition 4.5. The composition

φ′ ×_{N} φ′′ = φ′ ∘ φ′^{∗}(φ′′) = φ′′ ∘ φ′′^{∗}(φ′): N′ ×_{N} N′′ → N

is called the Whitney product of the coverings φ′ and φ′′.

Definition 4.6. A covering is said to be reducible, if it is equivalent to a covering of the form φ ×_{N} τ, where τ is a trivial covering. Otherwise it is called irreducible.

From now on, all coverings under consideration will be assumed to be smooth fiber bundles. The fiber dimension is called the dimension of the covering φ under consideration and is denoted by dim φ.

Proposition 4.1. Let E ⊂ J^{k}(π) be an equation in the bundle π: E → M and φ: N → E^{∞} be a smooth fiber bundle. Then the following statements are equivalent:

(1) The bundle φ is equipped with a structure of a covering.

(2) There exists a connection C^{φ} in the bundle π_{∞}∘φ: N → M, C^{φ}: X ↦ X^{φ}, X ∈ D(M), X^{φ} ∈ D(N), such that

(a) [X^{φ}, Y^{φ}] = [X, Y]^{φ}, i.e., C^{φ} is flat, and

(b) any vector field X^{φ} is projectible to E^{∞} under φ_{∗} and φ_{∗}(X^{φ}) =

CX, where C is the Cartan connection on E^{∞}.

The proof reduces to the check of definitions.

Using this result, we shall now obtain coordinate description of coverings. Namely, let x_{1}, …, x_{n}, u^{1}, …, u^{m} be local coordinates in J^{0}(π) and assume that internal coordinates in E^{∞} are chosen. Suppose also that over the neighborhood under consideration the bundle φ: N → E^{∞} is trivial with the fiber W and w^{1}, w^{2}, …, w^{s}, … are local coordinates in W. The functions w^{j} are called nonlocal coordinates in the covering φ. The connection C^{φ} puts into correspondence to any partial derivative ∂/∂x_{i} the vector field C^{φ}(∂/∂x_{i}) = D̃_{i}. By Proposition 4.1 on the preceding page, these vector fields are to be of the form

D̃_{i} = D_{i} + X_{i}^{v} = D_{i} + ∑_{α} X_{i}^{α} \frac{∂}{∂w^{α}} , i = 1, …, n,  (4.1)

where D_{i} are restrictions of total derivatives to E^{∞}, and satisfy the conditions

\begin{aligned}[D̃_{i}, D̃_{i}] = [D_{i}, D_{j}] + [D_{i}, X_{j}^{v}] & + [X_{i}^{v}, D_{j}] + [X_{i}^{v}, X_{j}^{v}] \\ & = [D_{i}, X_{j}^{v}] + [X_{i}^{v}, D_{j}] + [X_{i}^{v}, X_{j}^{v}] = 0 \end{aligned} (4.2)

for all i, j = 1, …, n.

We shall now prove a number of facts that simplify checking of triviality and equivalence of coverings.

Proposition 4.2. Let φ_{1} : N_{1} → E^{∞} and φ_{2} : N_{2} → E^{∞} be two coverings of the same dimension r < ∞. They are equivalent if and only if there exists a submanifold X ⊂ N_{1} ×_{E^{∞}} N_{2} such that

(1) codim X = r;

(2) The restrictions φ_{1}^{∗}(φ_{2}) |_{X} and φ_{2}^{∗}(φ_{1}) |_{X} are surjections.

(3) (D_{N_{1}×_{E^{∞}}N_{2}})_{θ} ⊂ T_{θ}X for any point θ ∈ X.

Proof. In fact, if ψ: N_{1} → N_{2} is an equivalence, then its graph G_{ψ} = { (y, ψ(y)) | y ∈ N_{1} } is the needed manifold X. Conversely, if X is a manifold satisfying (1)–(3), then the correspondence

y ↦ φ_{1}^{∗}(φ_{2})((φ_{1}^{∗}(φ_{2}))^{−1}(y) ∩ X)

is an equivalence.

Submanifolds X satisfying assumption (3) of the previous proposition are called invariant.

Proposition 4.3. Let φ_{1} : N_{1} → E^{∞} and φ_{2} : N_{2} → E^{∞} be two irreducible coverings of the same dimension r < ∞. Assume that the Whitney product of φ_{1} and φ_{2} is reducible and there exists an invariant submanifold X in N_{1} ×_{E^{∞}} N_{2} of codimension r. Then φ_{1} and φ_{2} are equivalent almost everywhere.

Proof. Since φ_{1}, φ_{2} are irreducible, X is to be mapped surjectively almost everywhere by φ_{1}^{∗}(φ_{2}) and φ_{2}^{∗}(φ_{1}) to N_{1} and N_{2} respectively (otherwise, their images would be invariant submanifolds). Hence, the coverings are equivalent by Proposition 4.2 on the preceding page.

Corollary 4.4. If φ_{1} and φ_{2} are one-dimensional coverings over E^{∞} and their Whitney product is reducible, then they are equivalent.

Proposition 4.5. Let φ: N → E^{∞} be a covering and U ⊂ E^{∞} be a domain such that the the manifold Ũ = φ^{−1}(U) is represented in the form U × ℝ^{r}, r ≤ ∞, while φ|Ũ is the projection to the first factor. Then the covering φ is locally irreducible if the system

D_{1}^{φ}(f) = 0, …, D_{n}^{φ}(f) = 0  (4.3)

has only constant solutions.

Proof. Suppose that there exists a solution f ≠ const of (4.3). Then, since the only solutions of the system

D_{1}(f) = 0, …, D_{n}(f) = 0,

where D_{i} is the restriction of the i-th total derivative to E^{∞}, are constants, f depends on one nonlocal variable w^{α} at least. Without loss of generality we may assume that ∂f/∂w^{1} ≠ 0 in a neighborhood U′ × V, U′ ⊂ U, V ⊂ ℝ^{r}. Define the diffeomorphism ψ: U′ ⊂ U → ψ(U′ ⊂ U) by setting

ψ(…, x_{i}, …, p_{σ}^{j}, …, w^{α}, …) = (…, x_{i}, …, p_{σ}^{j}, …, f, w^{2}, …, w^{α}, …).

Then ψ_{∗}(D_{i}^{φ}) = D_{i} + ∑_{α>1} X_{i}^{α}∂/∂w^{α} and consequently φ is reducible.

Let now φ be a reducible covering, i.e., φ = φ′ ×_{E^{∞}} τ, where τ is trivial. Then, if f is a smooth function on the total space of the covering τ, the function f^{∗} = (τ^{∗}(φ′))^{∗}(f) is a solution of (4.3). Obviously, there exists an f such that f^{∗} ≠ const.

4.2. Nonlocal symmetries and shadows. Let N be an object of Inf with the integrable distribution D = D_{N}. Define

D_{D}(N) = { X ∈ D(N) | [X, D] ⊂ D }

and set sym N = D_{D}(N)/D_{N}. Obviously, D_{D}(N) is a Lie ℝ-algebra and D is its ideal. Elements of the Lie algebra sym N are called symmetries of the object N.

Definition 4.7. Let φ: N → E^{∞} be a covering. A nonlocal φ-symmetry of E is an element of sym N. The Lie algebra of such symmetries is denoted by sym_{φ} E.

A base for computation of nonlocal symmetries is the following two results.

Theorem 4.6. Let φ: N → E^{∞} be a covering. The algebra sym_{φ} E is isomorphic to the Lie algebra of vector fields X on N such that

(1) X is vertical, i.e., X(φ^{∗}(f)) = 0 for any function f ∈ C^{∞}(M) ⊂ F(E);

(2) [X, D_{i}^{φ}] = 0, i = 1, …, n.

Proof. Note that the first condition means that in coordinate representation the coefficients of the field X at all ∂/∂x_{i} vanish. Hence the intersection of the set of vertical fields with D vanish. On the other hand, in any coset [X] ∈ sym_{φ} E there exists one and only one vertical element X^{v}. In fact, let X be an arbitrary element of [X]. Then X^{v} = X − ∑_{i} a_{i}D_{i}^{φ}, where a_{i} is the coefficient of X at ∂/∂x_{i}.

Theorem 4.7. Let φ: N = E^{∞} × ℝ^{r} → E^{∞} be the covering locally determined by the fields

D_{i}^{φ} = D_{i} + ∑_{α=1}^{r} X_{i}^{α} \frac{∂}{∂w^{α}} , i = 1, …, n, X_{i}^{α} ∈ F(N),

where w^{1}, w^{2}, … are coordinates in ℝ^{r} (nonlocal variables). Then any nonlocal φ-symmetry of the equation E = {F = 0} is of the form

˜�_{ψ,a} = ˜�_{ψ} + ∑_{α=1}^{r} a_{α} \frac{∂}{∂w^{α}} ,  (4.4)

where ψ = ψ^{1}, …, ψ^{m}, a = (a^{1}, …, a^{r}), ψ^{i}, a^{α} ∈ F(N) are functions satisfying the conditions

ℓ̃_{F} (ψ) = 0,  (4.5)
D̃_{i}(a^{α}) = ˜�_{ψ,a} (X_{i}^{α})  (4.6)

while

˜�_{ψ} = ∑_{j,σ} D̃_{σ}(ψ)\frac{∂}{∂u_{σ}^{j}}  (4.7)

and ℓ̃_{F} is obtained from ℓ_{F} by changing total derivatives D_{i} for D_{i}^{φ}.

Proof. Let X ∈ sym_{φ} E. Using Theorem 4.6, let us write down the field X in the form

X = ∑_{σ,j}′b_{σ}^{j} \frac{∂}{∂u_{σ}^{j}} + ∑_{α=1}^{r} a^{α} \frac{∂}{∂w^{α}} ,  (4.8)

where “prime” over the first sum means that the summation extends on internal coordinates in E^{∞} only. Then, equaling to zero the coefficient at ∂/∂u_{σ}^{j} in the commutator [X, D_{i}^{φ}], we obtain the following equations

D_{i}^{φ}(b_{σ}^{j}) = \{ \begin{matrix}b_{σi}^{j}, & if b_{σi}^{j} is an internal coordinate, \\ X(u_{σi}^{j}) & otherwise. \end{matrix}

Solving these equations, we obtain that the first summand in (4.8) on the preceding page is of the form ˜�_{ψ} , where ψ satisfies (4.5) on the facing page.

Comparing the result obtained with the description on local symmetries (see Theorem 3.29 on page 66), we see that in the nonlocal setting an additional obstruction arises represented by equation (4.6) on the facing page. Thus, in general, not every solution of (4.5) corresponds to a nonlocal φ-symmetry. We call vector fields ˜�_{ψ} of the form (4.7), where ψ satisfies equation (4.5), φ-shadows. In the next subsection it will be shown that for any φ-shadow ˜�_{ψ} there exists a covering φ′: N′ → N and a nonlocal φ ∘ φ′-symmetry S such that φ′_{∗}(S) = ˜�_{ψ} .

4.3. Reconstruction theorems. Let E ⊂ J^{k}(π) be a differential equation. Let us first establish relations between horizontal cohomology of E (see Definition 3.27 on page 59) and coverings over E^{∞}. All constructions below are realized in a local chart U ⊂ E^{∞}.

Consider a horizontal 1-form ω = ∑_{i=1}^{n} X_{i} dx_{i} ∈ Λ̄^{1}(E) and define on the space E^{∞} × ℝ the vector fields

D_{i}^{ω} = D_{i} + X_{i}∂/∂w, X_{i} ∈ F(E),  (4.9)

where w is a coordinate along ℝ. By direct computations, one can easily see that the conditions [D_{i}^{ω}, D_{j}^{ω}] = 0 fulfill if and only if d̄ω = 0. Thus, (4.9) determines a covering structure in the bundle φ : E^{∞} × ℝ → E^{∞} and this covering is denoted by φ^{ω}. It is also obvious that the covering φ^{ω} and φ^{ω′} are equivalent if and only if the forms ω and ω′ are cohomologous, i.e., if ω − ω′ = d̄f for some f ∈ F(E).

Let [ω_{1}], …, [ω^{α}], … be an ℝ-basis of the vector space H̄^{1}(E). Let us define the covering 𝔞_{1,0} : A^{1}(E) → E^{∞} as the Whitney product of all φ^{ω_{α}}. It can be shown that the equivalence class of 𝔞_{1,0} does not depend on the basis choice. Now, literary in the same manner as it was done in Definition 3.27 on page 59 for E^{∞}, we can define horizontal cohomology for A^{1}(E) and construct the covering 𝔞_{2,1} : A^{2}(E) → A^{1}(E), etc.

Definition 4.8. The inverse limit of the chain

⋯ → A^{k}(E) ⟶^{𝔞_{k,k−1}} A^{k−1}(E) → ⋯ → A^{1}(E) ⟶^{𝔞_{1,0}} E^{∞}  (4.10)

is called the universal Abelian covering of the equation E and is denoted by 𝔞 : A(E) → E^{∞}.

Obviously, H̄^{1}(A(E)) = 0.

Theorem 4.8 (see [21]). Let 𝔞 : A(E) → E^{∞} be the universal Abelian covering of the equation E = {F = 0}. Then any 𝔞-shadow reconstructs up to a nonlocal 𝔞-symmetry, i.e., for any solution ψ = (ψ^{1}, …, ψ^{m}), ψ^{j} ∈ F(A(E)), of the equation ℓ̃_{F} (ψ) = 0 there exists a set of functions a = (a_{α,i}), where a_{α,i} ∈ F(A(E)) such that ˜�_{ψ,a} is a nonlocal 𝔞-symmetry.

Proof. Let w^{j,α}, j ≤ k, be nonlocal variables in A^{k}(E) and assume that the covering structure in 𝔞 is determined by the vector fields D_{i}^{𝔞} = D_{i} + ∑_{j,α} X_{i}^{j,α}∂/∂w^{j,α}, where, by construction, X_{i}^{j,α} ∈ F(A^{j−1}(E)), i.e., the functions X_{i}^{j,α} do not depend on w^{k,α} for all k ≥ j.

Our aim is to prove that the system

D_{i}^{𝔞}(a_{j,α}) = ˜�_{ψ,a} (X_{i}^{j,α})  (4.11)

is solvable with respect to a = (a_{j,α}) for any ψ ∈ ker ℓ̃_{F}. We do this by induction on j. Note that

[D_{i}^{𝔞}, ˜�_{ψ,a} ] = ∑_{j,α} (D_{i}^{𝔞}(a_{j,α}) − ˜�_{ψ,a} (X_{i}^{j,α}))\frac{∂}{∂w^{j,α}}

for any set of functions (a_{j,α}). Then for j = 1 one has [D_{i}^{𝔞}, ˜�_{ψ,a} ](X_{k}^{1,α}) = 0, or

D_{i}^{𝔞}( ˜�_{ψ,a} (X_{k}^{1,α})) = ˜�_{ψ,a} (D_{i}^{𝔞}(X_{k}^{1,α})),

since X_{k}^{1,α} are functions on E^{∞}.

But from the construction of the covering 𝔞 one has D_{i}^{𝔞}(X_{k}^{1,α}) = D_{k}^{𝔞}(X_{i}^{1,α}), and we finally obtain

D_{i}^{𝔞}(�_{ψ}(X_{k}^{1,α})) = D_{k}^{𝔞}(�_{ψ}(X_{i}^{1,α})).

Note now that the equality H̄^{1}(A(E)) = 0 implies existence of functions a_{1,α} satisfying

D_{i}^{𝔞}(a_{1,α}) = �_{ψ}(X_{i}^{1,α}),

i.e., equation (4.11) is solvable for j = 1.

Assume now that solvability of (4.11) was proved for j < s and the functions (a_{1,α}, …, a_{j−1,α}) are some solutions. Then, since [D_{i}^{𝔞}, ˜�_{ψ,a} ] |_{A^{j−1}(E)} = 0, we obtain the needed a_{j,α} literally repeating the proof for the case j = 1.

Let now φ: N ⟶ E^{∞} be an arbitrary covering. The next result shows that any φ-shadow is reconstructable.

Theorem 4.9 (see also [22]). For any φ-shadow, i.e., for a solution ψ = (ψ^{1}, …, ψ^{m}), ψ^{j} ∈ F(N), of the equation ℓ̃_{F} (ψ) = 0, there exists a covering φ_{ψ}: N_{ψ} ⟶ N ⟶^{φ} E^{∞} and a φ_{ψ}-symmetry S_{ψ}, such that S_{ψ} |_{E^{∞}} = ˜�_{ψ} |_{E^{∞}}.

Proof. Let locally the covering φ be represented by the vector fields

D_{i}^{φ} = D_{i} + ∑_{α=1}^{r} X_{i}^{α} \frac{∂}{∂w^{α}} ,

r ≤ ∞ being the dimension of φ. Consider the space ℝ^{∞} with the coordinates w_{l}^{α}, α = 1, …, r, l = 0, 1, 2, …, w_{0}^{α} = w^{α}, and set N_{ψ} = N × ℝ^{∞} with

D_{i}^{φ_{ψ}} = D_{i} + ∑_{l,α} ( ˜�_{ψ} + S_{w})^{l} (X_{i}^{α})\frac{∂}{∂w_{l}^{α}} ,  (4.12)

where

˜�_{ψ} = ∑_{σ,k}′D_{σ}^{φ}(ψ^{k})\frac{∂}{∂u_{σ}^{k}} , S_{w} = ∑_{α,l} w_{l+1}^{α} \frac{∂}{∂w_{l}^{α}}  (4.13)

and “prime”, as before, denotes summation over internal coordinates. Set S_{ψ} = ˜�_{ψ} + S_{w}. Then

[S_{ψ}, D_{i}^{φ_{ψ}}] = ∑_{σ,k}′ ˜�_{ψ} (ū_{σi}^{k})\frac{∂}{∂u_{σ}^{k}} + ∑_{l,α} ( ˜�_{ψ} + S_{w})^{l+1} (X_{i}^{α})\frac{∂}{∂w_{l}^{α}}

− ∑_{σ,k}′D_{i}^{φ_{ψ}}(D_{σ}^{φ}(ψ^{k}))\frac{∂}{∂u_{σ}^{k}} − ∑_{l,α} ( ˜�_{ψ} + S_{w})^{l+1} (X_{i}^{α})\frac{∂}{∂w_{l}^{α}}

= ∑_{σ,k}′ ( ˜�_{ψ} (ū_{σi}^{k}) − D_{σi}^{φ}(ψ^{k})) \frac{∂}{∂u_{σ}^{k}} = 0.

Here, by definition, ū_{σi}^{k} = D_{i}^{φ}(u_{σ}^{k}) |_{N}.

Now, using the above proved equality, one has

[D_{i}^{φ_{ψ}}, D_{j}^{φ_{ψ}}] = ∑_{l,α} (D_{j}^{φ_{ψ}}( ˜�_{ψ} + S_{w})^{l}(X_{j}^{α}) − D_{j}^{φ_{ψ}}( ˜�_{ψ} + S_{w})^{l}(X_{i}^{α})) \frac{∂}{∂w_{l}^{α}}

= ∑_{l,α} ( ˜�_{ψ} + S_{w})^{l}(D_{i}^{φ_{ψ}}(X_{j}^{α}) − D_{j}^{φ_{ψ}}(X_{i}^{α}))\frac{∂}{∂w_{l}^{α}} = 0,

since D_{i}^{φ_{ψ}}(X_{j}^{α}) − D_{j}^{φ_{ψ}}(X_{i}^{α}) = D_{i}^{φ}(X_{j}^{α}) − D_{j}^{φ}(X_{i}^{α}) = 0.

Let now φ: N ⟶ E^{∞} be a covering and φ′: N′ ⟶ N ⟶^{φ} E^{∞} be another one. Then, by obvious reasons, any φ-shadow ψ is a φ′-shadow as well. Applying the construction of Theorem 4.9 to both φ and φ′, we obtain two coverings, φ_{ψ} and φ′_{ψ} respectively.

Lemma 4.10. The following commutative diagram of coverings

N′_{ψ} ⟶ N_{ψ}
↓  ↓
N′ ⟶ N ⟶ E^{∞}

takes place. Moreover, if S_{ψ} and S′_{ψ} are nonlocal symmetries corresponding in N_{ψ} and N′_{ψ} constructed by Theorem 4.9 on the page before, then S′_{ψ} |_{F(N_{ψ})} = S_{ψ}.

Proof. It suffices to compare expressions (4.12) and (4.13) on the preceding page for the coverings N_{ψ} and N′_{ψ}.

As a corollary of Theorem 4.9 and of the previous lemma, we obtain the following result.

Theorem 4.11. Let φ: N ⟶ E^{∞}, E = { F = 0 }, be an arbitrary covering and ψ_{1}, …, ψ_{s} ∈ F(N), be solutions of the equation ℓ̃_{F} (ψ) = 0. Then there exists a covering φ_{Ψ} : N_{Ψ} ⟶ N ⟶^{φ} E^{∞} and φ_{Ψ}-symmetries S_{ψ_{1}}, …, S_{ψ_{s}}, such that S_{ψ_{s}} |_{E^{∞}} = ˜�_{ψ_{i}} |_{E^{∞}}, i = 1, …, s.

Proof. Consider the section ψ_{1} and the covering φ_{ψ_{1}} : N_{ψ_{1}} ⟶^{φ̄_{ψ_{1}}} N ⟶^{φ} E^{∞} together with the symmetry S_{ψ_{1}} constructed in Theorem 4.9 on the page before. Then ψ_{2} is a φ_{ψ_{1}}-shadow and we can construct the covering

φ_{ψ_{1},ψ_{2}} : N_{ψ_{1},ψ_{2}} ⟶^{φ̄_{ψ_{1},ψ_{2}}} N_{ψ_{1}} ⟶^{φ_{ψ_{1}}} E^{∞} with the symmetry S_{ψ_{2}}. Applying this

procedure step by step, we obtain the series of coverings

N_{ψ_{1},...,ψ_{s}} ⟶^{φ̄_{ψ_{1},...,ψs}} N_{ψ_{1},...,ψ_{s−1}} ⟶^{φ̄_{ψ_{1},...,ψ_{s−1}}} ⋯ ⟶^{φ̄_{ψ_{1},ψ_{2}}} N_{ψ_{1}} ⟶^{φ̄_{ψ_{1}}} N ⟶^{φ} E^{∞}.

with the symmetries S_{ψ_{1}}, …, S_{ψ_{s}}. But ψ_{1} is a φ_{ψ_{1},...,ψ_{s}}-shadow and we can construct the covering φ_{ψ_{1}} : N_{ψ_{1}}^{(1)} ⟶ N_{ψ_{1},...,ψ_{s}} ⟶ E^{∞} with the symmetry S_{ψ_{1}}^{(1)}

satisfying S_{ψ_{1}}^{(1)} |_{F(N_{ψ_{1}})} = S_{ψ_{1}} (see Lemma 4.10), etc. Passing to the inverse limit, we obtain the covering N_{Ψ} we need.

5. Frölicher–Nijenhuis brackets and recursion operators

We return back to the general algebraic setting of Section 1 and extend standard constructions of calculus to form-valued derivations. It allows us to define Frölicher–Nijenhuis brackets and introduce a cohomology theory (C-cohomologies) associated to commutative algebras with flat connections. Applying this theory to partial differential equations, we obtain an algebraic description of recursion operators for symmetries and describe efficient tools to compute these operators. For technical details, examples and generalizations and we refer the reader to the papers [24, 28, 27] and [29, 31, 30].

In Subsection 6.4, C-cohomologies will be discussed again in the general framework of horizontal cohomologies with coefficients.

5.1. Calculus in form-valued derivations. Let 𝕜 be a field of characteristic zero and A be a commutative unitary 𝕜-algebra. Let us recall the basic notations from Section 1:

• D(P) is the module of P -valued derivations A ⟶ P, where P is an A-module;

• D_{i}(P) is the module of P -valued skew-symmetric i-derivations. In particular, D_{1}(P) = D(P);

• Λ^{i}(A) is the module of differential i-forms of the algebra A;

• d: Λ^{i}(A) ⟶ Λ^{i+1}(A) is the de Rham differential. Recall also that the modules Λ^{i}(A) are representative objects for the functors D_{i}: P ⇒ D_{i}(P), i.e., D_{i}(P) = Hom_{A}(Λ^{i}(A), P). The isomorphism D(P) = Hom_{A}(Λ^{1}(A), P) can be expressed in more exact terms: for any derivation X: A ⟶ P, there exists a uniquely defined homomorphism φ^{X}: Λ^{1}(A) ⟶ P satisfying X = φ^{X} ∘ d. Denote by ⟨Z, ω⟩ ∈ P the value of the derivation Z ∈ D_{i}(P) at ω ∈ Λ^{i}(A).

Both Λ^{∗}(A) = ⨁_{i≥0} Λ^{i}(A) and D_{∗}(A) = ⨁_{i≥0} D_{i}(A) are endowed with the structures of superalgebras with respect to the wedge product operation ∧: Λ^{i}(A) ⊗ Λ^{j}(A) ⟶ Λ^{i+j}(A) and ∧ : D_{i}(A) ⊗ D_{j}(A) ⟶ D_{i+j}(A), the de Rham differential d: Λ^{∗}(A) ⟶ Λ^{∗}(A) becoming a derivation of Λ^{∗}(A). Note also that D_{∗}(P) = ⨁_{i≥0} D_{i}(P) is a D_{∗}(A)-module.

Using the paring ⟨·, ·⟩ and the wedge product, we define the inner product (or contraction) i_{X}ω ∈ Λ^{j−i}(A) of X ∈ D_{i}(A) and ω ∈ Λ^{j}(A), i ≤ j, by setting

⟨Y, i_{X}ω⟩ = (−1)^{i(j−i)}⟨X ∧ Y, ω⟩,  (5.1)

where Y is an arbitrary element of D_{j−i}(P), P being an A-module. We formally set i_{X}ω = 0 for i > j. When i = 1, this definition coincides with the one given in Section 1. Recall that the following duality is valid:

⟨X, da ∧ ω⟩ = ⟨X(a), ω⟩,  (5.2)

where ω ∈ Λ^{i}(A), X ∈ D_{i+1}(P), and a ∈ A (see Exercise 1.4 on page 14). Using the property (5.2), one can show that

i_{X}(ω ∧ θ) = i_{X}(ω) ∧ θ + (−1)^{Xω}ω ∧ i_{X}(ω)

for any ω, θ ∈ Λ^{∗}(A), where (as everywhere below) the symbol of a graded object used as the exponent of (−1) denotes the degree of that object.

We now define the Lie derivative of ω ∈ Λ^{∗}(A) along X ∈ D_{∗}(A) as

L_{X}ω = (i_{X} ∘ d − (−1)^{X}d ∘ i_{X})ω = [i_{X}, d]ω,  (5.3)

where [·, ·] denotes the supercommutator: if Δ, Δ′ : Λ^{∗}(A) ⟶ Λ^{∗}(A) are graded derivations, then [Δ, Δ′] = Δ ∘ Δ′ − (−1)^{ΔΔ′}Δ′ ∘ Δ. For X ∈ D(A) this definition coincides with the one given by equality (1.9) on page 15.

Consider now the graded module D(Λ^{∗}(A)) of Λ^{∗}(A)-valued derivations A ⟶ Λ^{∗}(A) (corresponding to form-valued vector fields—or, which is the same—vector-valued differential forms on a smooth manifold). Note that the graded structure in D(Λ^{∗}(A)) is determined by the splitting D(Λ^{∗}(A)) = ⨁_{i≥0} D(Λ^{i}(A)) and thus elements of grading i are derivations X such that im X ⊂ Λ^{i}(A). We shall need three algebraic structures associated to D(Λ^{∗}(A)). First note that D(Λ^{∗}(A)) is a graded Λ^{∗}(A)-module: for any X ∈ D(Λ^{∗}(A)), ω ∈ Λ^{∗}(A) and a ∈ A we set (ω ∧ X)a = ω ∧ X(a). Second, we can define the inner product i_{X}ω ∈ Λ^{i+j−1}(A) of X ∈ D(Λ^{i}(A)) and ω ∈ Λ^{j}(A) in the following way. If j = 0, we set i_{X}ω = 0. Then, by induction on j and using the fact that Λ^{∗}(A) as a graded A-algebra is generated by the elements of the form da, a ∈ A, we set

i_{X}(da ∧ ω) = X(a) ∧ ω − (−1)^{X}da ∧ i_{X}(ω), a ∈ A.  (5.4)

Finally, we can contract elements of D(Λ^{∗}(A)) with each other in the following way:

(i_{X}Y)a = i_{X}(Y a), X, Y ∈ D(Λ^{∗}(A)), a ∈ A.  (5.5)

Three properties of contractions are essential in the sequel.

Proposition 5.1. Let X, Y ∈ D(Λ^{∗}(A)) and ω, θ ∈ Λ^{∗}(A). Then

i_{X}(ω ∧ θ) = i_{X}(ω) ∧ θ + (−1)^{ω(X−1)}ω ∧ i_{X}(θ),  (5.6)
i_{X}(ω ∧ Y) = i_{X}(ω) ∧ Y + (−1)^{ω(X−1)}ω ∧ i_{X}(Y),  (5.7)
[i_{X}, i_{Y}] = i_{[[X,Y]]^{rn}},  (5.8)

where

[[X, Y]]^{rn} = i_{X}(Y) − (−1)^{(X−1)(Y −1)}i_{Y} (X).  (5.9)

Proof. Equality (5.6) is a direct consequence of (5.4). To prove (5.7), it suffices to use the definition and expressions (5.5) and (5.6).

Let us prove (5.8) now. To do this, note first that due to (5.5) the equality is sufficient to be checked on elements ω ∈ Λ^{j}(A). Let us use induction on j. For j = 0 it holds in a trivial way. Let a ∈ A; then one has

[i_{X}, i_{Y}](da ∧ ω) = (i_{X} ∘ i_{Y} − (−1)^{(X−1)(Y −1)}i_{Y} ∘ i_{X})(da ∧ ω)

= i_{X}(i_{Y} (da ∧ ω)) − (−1)^{(X−1)(Y −1)}i_{Y} (i_{X}(da ∧ ω)).

But

i_{X}(i_{Y} (da ∧ ω)) = i_{X}(Y (a) ∧ ω − (−1)^{Y} da ∧ i_{Y} ω)

= i_{X}(Y (a)) ∧ ω + (−1)^{(X−1)Y} Y (a) ∧ i_{X}ω − (−1)^{Y} (X(a) ∧ i_{Y} ω
− (−1)^{X}da ∧ i_{X}(i_{Y} ω)),

while

i_{Y} (i_{X}(da ∧ ω) = i_{Y} (X(a) ∧ ω − (−1)^{X}da ∧ i_{X}ω)

= i_{Y} (X(a)) ∧ ω + (−1)^{X(Y −1)}X(a) ∧ i_{Y} ω − (−1)^{X}(Y (a) ∧ ω
− (−1)^{Y} da ∧ i_{Y} (i_{X}ω)).

Hence,

[i_{X}, i_{Y}](da ∧ ω) = (i_{X}(Y (a)) − (−1)^{(X−1)(Y −1)}i_{Y} (X(a))) ∧ ω

+ (−1)^{X+Y} da ∧ (i_{X}(i_{Y} ω) − (−1)^{(X−1)(Y −1)}i_{Y} (i_{X}ω)).

But, by definition,

i_{X}(Y (a)) − (−1)^{(X−1)(Y −1)}i_{Y} (X(a))

= (i_{X}Y − (−1)^{(X−1)(Y −1)}i_{Y} X)(a) = [[X, Y]]^{rn}(a),

whereas

i_{X}(i_{Y} ω) − (−1)^{(X−1)(Y −1)}i_{Y} (i_{X}ω) = i_{[[X,Y]]^{rn}}(ω)

by induction hypothesis.

Definition 5.1. The element [[X, Y]]^{rn} defined by equality (5.9) is called the Richardson–Nijenhuis bracket of elements X and Y.

Directly from Proposition 5.1 we obtain the following

Proposition 5.2. For any derivations X, Y, Z ∈ D(Λ^{∗}(A)) and a form ω ∈ Λ^{∗}(A) one has

[[X, Y]]^{rn} + (−1)^{(X+1)(Y +1)}[[Y, X]]^{rn} = 0,  (5.10)
∮ (−1)^{(Y +1)(X+Z)}[[[[X, Y]]^{rn}, Z]]^{rn} = 0,  (5.11)
[[X, ω ∧ Y]]^{rn} = i_{X}(ω) ∧ Y + (−1)^{(X+1)ω}ω ∧ [[X, Y]]^{rn}.  (5.12)

Here and below the symbol ∮ denotes the sum of cyclic permutations.

Remark 5.1. Note that Proposition 5.2 means that D(Λ^{∗}(A))^{↓} is a Gerstenhaber algebra with respect to the Richardson–Nijenhuis bracket [23]. Here the superscript ↓ denotes the shift of grading by 1.

Similarly to (5.3) define the Lie derivative of ω ∈ Λ^{∗}(A) along X ∈ D(Λ^{∗}(A)) by

L_{X}ω = (i_{X} ∘ d + (−1)^{X}d ∘ i_{X})ω = [i_{X}, d]ω  (5.13)

(the change of sign is due to the fact that deg(i_{X}) = deg(X) − 1). From the properties of i_{X} and d we obtain

Proposition 5.3. For any X ∈ D(Λ^{∗}(A)) and ω, θ ∈ Λ^{∗}(A), one has the following identities:

L_{X}(ω ∧ θ) = L_{X}(ω) ∧ θ + (−1)^{Xω}ω ∧ L_{X}(θ),  (5.14)
L_{ω∧X} = ω ∧ L_{X} + (−1)^{ω+X}d(ω) ∧ i_{X},  (5.15)
[L_{X}, d] = 0.  (5.16)

Our main concern now is to analyze the commutator [L_{X}, L_{Y}] of two Lie derivatives. It may be done efficiently for smooth algebras (see Definition 1.9 on page 18).

Proposition 5.4. Let A be a smooth algebra. Then for any derivations X, Y ∈ D(Λ^{∗}(A)) there exists a uniquely determined element [[X, Y]]^{fn} ∈ D(Λ^{∗}(A)) such that

[L_{X}, L_{Y}] = L_{[[X,Y]]^{fn}}.  (5.17)

Proof. To prove existence, recall that for smooth algebras one has

D_{i}(P) = Hom_{A}(Λ^{i}(A), P) = P ⊗_{A} Hom_{A}(Λ^{i}(A), A) = P ⊗_{A} D_{i}(A)

for any A-module P and integer i ≥ 0. Using this identification, represent elements X, Y ∈ D(Λ^{∗}(A)) in the form

X = ω ⊗ X′ and Y = θ ⊗ Y′ for ω, θ ∈ Λ^{∗}(A), X′, Y′ ∈ D(A).

Then it is easily checked that the element

Z = ω ∧ θ ⊗ [X′, Y′] + ω ∧ L_{X′}θ ⊗ Y + (−1)^{ω}dω ∧ i_{X′}θ ⊗ Y′
− (−1)^{ωθ}θ ∧ L_{Y′}ω ⊗ X′ − (−1)^{(ω+1)θ}dθ ∧ i_{Y′}ω ⊗ X′  (5.18)
= ω ∧ θ ⊗ [X′, Y′] + L_{X}θ ⊗ Y′ − (−1)^{ωθ}L_{Y} ω ⊗ X′

satisfies (5.17).

Uniqueness follows from the fact that L_{X}(a) = X(a) for any a ∈ A.

Definition 5.2. The element [[X, Y]]^{fn} ∈ D^{i+j}(Λ^{∗}(A)) defined by formula (5.17) is called the Frölicher–Nijenhuis bracket of elements X ∈ D^{i}(Λ^{∗}(A)) and Y ∈ D^{j}(Λ^{∗}(A)).

The basic properties of this bracket are summarized in the following

Proposition 5.5. Let A be a smooth algebra, X, Y, Z ∈ D(Λ^{∗}(A)) be derivations and ω ∈ Λ^{∗}(A) be a differential form. Then the following identities are valid:

[[X, Y]]^{fn} + (−1)^{XY} [[Y, X]]^{fn} = 0,  (5.19)
∮ (−1)^{Y (X+Z)}[[X, [[Y, Z]]^{fn}]]^{fn} = 0,  (5.20)
i_{[[X,Y]]^{fn}} = [L_{X}, i_{Y}] + (−1)^{X(Y +1)}L_{i_{Y} X},  (5.21)
i_{Z}[[X, Y]]^{fn} = [[i_{Z}X, Y]]^{fn} + (−1)^{X(Z+1)}[[X, i_{Z}Y]]^{fn}

+ (−1)^{X}i_{[[Z,X]]^{fn}}Y − (−1)^{(X+1)Y} i_{[[Z,Y]]^{fn}}X, (5.22)

\begin{aligned}[[X, ω ∧ Y]]^{fn} = L_{X}ω ∧ Y − (−1)^{(X+1)(Y +ω)} & dω ∧ i_{Y} X \\ & + (−1)^{Xω}ω ∧ [[X, Y]]^{fn}. \end{aligned} (5.23)

Note that the first two equalities in the previous proposition mean that the module D(Λ^{∗}(A)) is a Lie superalgebra with respect to the Frölicher– Nijenhuis bracket.

Remark 5.2. The above exposed algebraic scheme has a geometrical realization, if one takes A = C^{∞}(M), M being a smooth finite-dimensional manifold. The algebra A = C^{∞}(M) is smooth in this case. However, in the geometrical theory of differential equations we have to work with infinite-dimensional manifolds^{10} of the form N = proj lim_{{π_{k+1,k}}} N_{k}, where all the maps π_{k+1,k}: N_{k+1} ⟶ N_{k} are surjections of finite-dimensional smooth manifolds. The corresponding algebraic object is a filtered algebra A = ⋃_{k∈ℤ} A_{k}, A_{k} ⊂ A_{k+1}, where all A_{k} are subalgebras in A. As it was already noted, self-contained differential calculus over A is constructed, if one considers the category of all filtered A-modules with filtered homomorphisms for morphisms between them. Then all functors of differential calculus in this category become filtered, as well as their representative objects.

In particular, the A-modules Λ^{i}(A) are filtered by A_{k}-modules Λ^{i}(A_{k}). We say that the algebra A is finitely smooth, if Λ^{1}(A_{k}) is a projective A_{k}- module of finite type for any k ∈ ℤ. For finitely smooth algebras, elements of D(P) may be represented as formal infinite sums ∑_{k} p_{k} ⊗ X_{k}, such that any finite sum S_{n} = ∑_{k≤n} p_{k} ⊗X_{k} is a derivation A_{n} ⟶ P_{n+s} for some fixed s ∈ ℤ. Any derivation X is completely determined by the system {S_{n}} and Proposition 5.5 obviously remains valid.

10Infinite jets, infinite prolongations of differential equations, total spaces of coverings, etc.

5.2. Algebras with flat connections and cohomology. We now introduce the second object of our interest. Let A be an 𝕜-algebra, 𝕜 being a field of zero characteristic, and B be an algebra over A. We shall assume that the corresponding homomorphism φ: A ⟶ B is an embedding. Let P be a B-module; then it is an A-module as well and we can consider the B-module D(A, P) of P -valued derivations A ⟶ P.

Definition 5.3. Let ∇^{•}: D(A, ·) ⇒ D(·) be a natural transformations of functors D(A, ·): A ⇒ D(A, P) and D(·): P ⇒ D(·) in the category of Bmodules, i.e., a system of homomorphisms ∇^{P} : D(A, P) ⟶ D(P) such that the diagram

D(A, P) ⟶^{∇^{P}} D(P)
D(A,f)↓  ↓^{D(f)}
D(A, Q) ⟶^{∇^{Q}} D(Q)

is commutative for any B-homomorphism f: P ⟶ Q. We say that ∇^{•} is a connection in the triad (A, B, φ), if ∇^{P} (X) |_{A} = X for any X ∈ D(A, P).

Here and below we use the notation Y |_{A} = Y ∘ φ for any Y ∈ D(P).

Remark 5.3. When A = C^{∞}(M), B = C^{∞}(E), φ = π^{∗}, where M and E are smooth manifolds and π: E ⟶ M is a smooth fiber bundle, Definition 5.3 reduces to the ordinary definition of a connection in the bundle π. In fact, if we have a connection ∇^{•} in the sense of Definition 5.3, then the correspondence

D(A) ↪ D(A, B) ⟶^{∇^{B}} D(B)

allows one to lift any vector field on M up to a π-projectible field on E. Conversely, if ∇ is such a correspondence, then we can construct a natural transformation ∇^{•} of the functors D(A, ·) and D(·) due to the fact that for smooth finite-dimensional manifolds one has D(A, P) = P ⊗_{A} D(A) and D(P) = P ⊗_{B} D(P) for an arbitrary B-module P. We use the notation ∇ = ∇^{B} in the sequel.

Definition 5.4. Let ∇^{•} be a connection in (A, B, φ) and X, Y ∈ D(A, B) be two derivations. The curvature form of the connection ∇^{•} on the pair X, Y is defined by

R_{∇}(X, Y) = [∇(X), ∇(Y)] − ∇(∇(X) ∘ Y − ∇(Y) ∘ X).  (5.24)

Note that (5.24) makes sense, since ∇(X) ∘ Y − ∇(Y) ∘ X is a B-valued derivation of A.

Consider now the de Rham differential d = d_{B}: B ⟶ Λ^{1}(B). Then the composition d_{B} ∘ φ: A ⟶ B is a derivation. Consequently, we may consider the derivation ∇(d_{B} ∘ φ) ∈ D(Λ^{1}(B)).

Definition 5.5. The element U_{∇} ∈ D(Λ^{1}(B)) defined by

U_{∇} = ∇(d_{B} ∘ φ) − d_{B}  (5.25)

is called the connection form of ∇.

Directly from the definition we obtain the following Lemma 5.6. The equality

i_{X}(U_{∇}) = X − ∇(X |_{A})  (5.26)

holds for any X ∈ D(B).

Using this result, we now prove Proposition 5.7. If B is a smooth algebra, then

i_{Y} i_{X}[[U_{∇}, U_{∇}]]^{fn} = 2R_{∇}(X |_{A} , Y |_{A})  (5.27)

for any X, Y ∈ D(B). Proof. First note that deg U_{∇} = 1. Then using (5.22) and (5.19) we obtain

i_{X}[[U_{∇}, U_{∇}]]^{fn} = [[i_{X}U_{∇}, U_{∇}]]^{fn} + [[U_{∇}, i_{X}U_{∇}]]^{fn} − i_{[[X,U_{∇}]]^{fn}}U_{∇} − i_{[[X,U_{∇}]]^{fn}}U_{∇}

= 2([[i_{X}U_{∇}, U_{∇}]]^{fn} − i_{[[X,U_{∇}]]^{fn}}U_{∇}).

Applying i_{Y} to the last expression and using (5.20) and (5.22), we get now

i_{Y} i_{X}[[U_{∇}, U_{∇}]]^{fn} = 2([[i_{X}U_{∇}, i_{Y} U_{∇}]]^{fn} − i_{[[X,Y]]^{fn}}U_{∇}).

But [[V, W]]^{fn} = [V, W] for any V, W ∈ D(Λ^{0}(A)) = D(A). Hence, by (5.26), we have i_{Y} i_{X}[[U_{∇}, U_{∇}]]^{fn} = 2([X − ∇(X |_{A}), Y − ∇(Y |_{A})] − ([X, Y] − ∇([X, Y] |_{A}))). It only remains to note now that ∇(X |_{A}) |_{A} = X |_{A} and [X, Y] |_{A} = X ∘ Y |_{A} − Y ∘ X |_{A}. Definition 5.6. A connection ∇ in (A, B, φ) is called flat, if R_{∇} = 0. Thus for flat connections we have

[[U_{∇}, U_{∇}]]^{fn} = 0.  (5.28)

Let U ∈ D(Λ^{1}(B)) be an element satisfying (5.28). Then from the graded Jacobi identity (5.20) we obtain 2[[U, [[U, X]]^{fn}]]^{fn} = [[[[U, U]]^{fn}, X]]^{fn} = 0 for any X ∈ D(Λ^{∗}(A)). Consequently, the operator ∂_{U} = [[U, ·]]^{fn} : D(Λ^{i}(B)) ⟶ D(Λ^{i+1}(B)) defined by the equality ∂_{U}(X) = [[U, X]]^{fn} satisfies the identity ∂_{U} ∘ ∂_{U} = 0.

Consider now the case U = U_{∇}, where ∇ is a flat connection. Definition 5.7. An element X ∈ D(Λ^{∗}(B)) is called vertical, if X(a) = 0 for any a ∈ A. Denote the B-submodule of such elements by D^{v}(Λ^{∗}(B)).

Lemma 5.8. Let ∇ be a connection in (A, B, φ). Then (1) an element X ∈ D(Λ^{∗}(B)) is vertical if and only if i_{X}U_{∇} = X;

(2) the connection form U_{∇} is vertical, U_{∇} ∈ D^{v}(Λ^{1}(B));

(3) the map ∂_{U_{∇}} preserves verticality, ∂_{U_{∇}}(D^{v}(Λ^{i}(B))) ⊂ D^{v}(Λ^{i+1}(B)).

Proof. To prove (1), use Lemma 5.6: from (5.26) it follows that i_{X}U_{∇} = X if and only if ∇(X |_{A}) = 0. But ∇(X |_{A}) |_{A} = X |_{A}. The second statements follows from the same lemma and from the first one:

i_{U_{∇}}U_{∇} = U_{∇} − ∇(U_{∇} |_{A}) = U_{∇} − ∇((U_{∇} − ∇(U_{∇} |_{A}) |_{A} ) = U_{∇}.

Finally, (3) is a consequence of (5.22).

Definition 5.8. Denote the restriction ∂_{U_{∇}} |_{D^{v}(Λ^{∗}(A))} by ∂_{∇} and call the complex

0 ⟶ D^{v}(B) ⟶^{∂_{∇}} D^{v}(Λ^{1}(B)) ⟶ ⋯ ⟶ D^{v}(Λ^{i}(B)) ⟶^{∂_{∇}} D^{v}(Λ^{i+1}(B)) ⟶ ⋯ (5.29)

the ∇-complex of the triple (A, B, φ). The corresponding cohomology is denoted by H_{∇}^{∗}(B; A, φ) = ⨁_{i≥0} H_{∇}^{i}(B; A, φ) and is called the ∇-cohomology of the triple (A, B, φ).

Introduce the notation

d_{∇}^{v} = L_{U_{∇}}: Λ^{i}(B) ⟶ Λ^{i+1}(B).  (5.30)

Proposition 5.9. Let ∇ be a flat connection in the triple (A, B, φ) and B be a smooth (or finitely smooth) algebra. Then for any X, Y ∈ D^{v}(Λ^{∗}(A)) and ω ∈ Λ^{∗}(A) one has

∂_{∇}[[X, Y]]^{fn} = [[∂_{∇}X, Y]]^{fn} + (−1)^{X}[[X, ∂_{∇}Y]]^{fn},  (5.31)
[i_{X}, ∂_{∇}] = (−1)^{X}i_{∂_{∇}X},  (5.32)
∂_{∇}(ω ∧ X) = (d_{∇}^{v} − d)(ω) ∧ X + (−1)^{ω}ω ∧ ∂_{∇}X,  (5.33)
[d_{∇}^{v}, i_{X}] = i_{∂_{∇}X} + (−1)^{X}L_{X}.  (5.34)

Proof. Equality (5.31) is a direct consequence of (5.20). Equality (5.32) follows from (5.22). Equality (5.33) follows from (5.23) and (5.26). Finally, (5.34) is obtained from (5.21).

Corollary 5.10. The cohomology module H_{∇}^{∗}(B; A, φ) inherits the graded Lie algebra structure with respect to the Frölicher–Nijenhuis bracket [[·, ·]]^{fn}, as well as to the contraction operation.

Proof. Note that D^{v}(Λ^{∗}(A)) is closed with respect to the Frölicher–Nijenhuis bracket: to prove this fact, it suffices to apply (5.22). Then the first statement follows from (5.31). The second one is a consequence of (5.32).

Remark 5.4. We preserve the same notations for the inherited structures. Note, in particular, that H_{∇}^{0}(B; A, φ) is a Lie algebra with respect to the Frölicher–Nijenhuis bracket (which reduces to the ordinary Lie bracket in this case). Moreover, H_{∇}^{1}(B; A, φ) is an associative algebra with respect to the inherited contraction, while the action

R_{Ω} : X ↦ i_{X}Ω, X ∈ H_{∇}^{0}(B; A, φ), Ω ∈ H_{∇}^{1}(B; A, φ)

is a representation of this algebra as endomorphisms of H_{∇}^{0}(B; A, φ).

Consider now the map d_{∇}^{v}: Λ^{∗}(B) ⟶ Λ^{∗}(B) defined by (5.30) and define d_{∇}^{h} = d_{B} − d_{∇}^{v}.

Proposition 5.11. Let B be a (finitely) smooth algebra and ∇ be a smooth connection in the triple (B; A, φ). Then

(1) The pair (d_{∇}^{h}, d_{∇}^{v}) forms a bicomplex, i.e.

d_{∇}^{v} ∘ d_{∇}^{v} = 0, d_{∇}^{h} ∘ d_{∇}^{h} = 0, d_{∇}^{h} ∘ d_{∇}^{v} + d_{∇}^{v} ∘ d_{∇}^{h} = 0.  (5.35)

(2) The differential d_{∇}^{h} possesses the following properties

[d_{∇}^{h}, i_{X}] = −i_{∂_{∇}X},  (5.36)
∂_{∇}(ω ∧ X) = −d_{∇}^{h}(ω) ∧ X + (−1)^{ω}ω ∧ ∂_{∇}X,  (5.37)

where ω ∈ Λ^{∗}(B), X ∈ D^{v}(Λ^{∗}(B)).

Proof. (1) Since deg d_{∇}^{v} = 1, we have

2d_{∇}^{v} ∘ d_{∇}^{v} = [d_{∇}^{v}, d_{∇}^{v}] = [L_{U_{∇}}, L_{U_{∇}}] = L_{[[U_{∇},U_{∇}]]^{fn}} = 0.

Since d_{∇}^{v} = L_{U_{∇}}, the identity [d_{B}, d_{∇}^{v}] = 0 holds (see (5.16)), and it concludes the proof of the first part.

(2) To prove (5.36), note that

[d_{∇}^{h}, i_{X}] = [d_{B} − d_{∇}^{h}, i_{X}] = (−1)^{X}L_{X} − [d_{∇}^{v}, i_{X}],

and (5.36) holds due to (5.34). Finally, (5.37) is just the other form of (5.33).

Definition 5.9. Let ∇ be a connection in (A, B, φ).

(1) The bicomplex (B, d_{∇}^{h}, d_{∇}^{v}) is called the variational bicomplex associated to the connection ∇.

(2) The corresponding spectral sequence is called the ∇-spectral sequence of the triple (A, B, φ).

Obviously, the ∇-spectral sequence converges to the de Rham cohomology of B.

To finish this section, note the following. Since the module Λ^{1}(B) is generated by the image of the operator d_{B}: B ⟶ Λ^{1}(B) while the graded algebra Λ^{∗}(B) is generated by Λ^{1}(B), we have the direct sum decomposition

Λ^{∗}(B) = ⨁_{i≥0} ⨁_{p+q=i} Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B),

where

Λ_{v}^{p}(B) = \underbrace{Λ_{v}^{1}(B) ∧ ⋯ ∧ Λ_{v}^{1}(B)}_{p times}, Λ_{h}^{q}(B) = \underbrace{Λ_{h}^{1}(B) ∧ ⋯ ∧ Λ_{h}^{1}(B)}_{q times},

while the submodules Λ_{v}^{1}(B) ⊂ Λ^{1}(B), Λ_{h}^{1}(B) ⊂ Λ^{1}(B) are spanned in Λ^{1}(B) by the images of the differentials d_{∇}^{v} and d_{∇}^{h} respectively. Obviously, we have the following embeddings:

d_{∇}^{h} (Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B)) ⊂ Λ_{v}^{p}(B) ⊗ Λ_{h}^{q+1}(B),
d_{∇}^{v} (Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B)) ⊂ Λ_{v}^{p+1}(B) ⊗ Λ_{h}^{q}(B).

Denote by D^{p,q}(B) the module D^{v}(Λ_{v}^{p}(B) ⊗ Λ_{h}^{q}(B)). Then, obviously, D^{v}(B) = ⨁_{i≥0} ⨁_{p+q=i} D^{p,q}(B), while from equalities (5.36) and (5.37) we obtain

∂_{∇}(D^{p,q}(B)) ⊂ D^{p,q+1}(B).

Consequently, the module H_{∇}^{∗}(B; A, φ) is split as

H_{∇}^{∗}(B; A, φ) = ⨁_{i≥0} ⨁_{p+q=i} H_{∇}^{p,q}(B; A, φ)  (5.38)

with the obvious meaning of the notation H_{∇}^{p,q}(B; A, φ).

5.3. Applications to differential equations: recursion operators. Now we apply the above exposed algebraic results to the case of infinitely prolonged differential equations. Let us start with establishing a correspondence between geometric constructions of Section 3 and algebraic ones presented in the previous two subsections.

Let E ⊂ J^{k}(π) be a formally integrable equation (see Definition 3.20 on page 54) and E^{∞} ⊂ J^{∞}(π) be its infinite prolongations. Then the bundle π_{∞} : E^{∞} ⟶ M is endowed with the Cartan connection C (Definition 3.23 on page 56) and this connection is flat (Corollary 3.19 on page 57). Thus the triple

(A = C^{∞}(M), B = F(E), φ = π_{∞}^{∗})

with ∇ = C is an algebra with a flat connection, A being a smooth and B being a finitely smooth algebra. The corresponding connection form is exactly the structural element U_{C} of the equation E (see Definition 3.24 on page 57).

Thus, to any formally integrable equation E ⊂ J^{k}(π) we can associate the complex

0 ⟶ D^{v}(E) ⟶^{∂_{C}} D^{v}(Λ^{1}(E)) ⟶ ⋯ ⟶ D^{v}(Λ^{i}(E)) ⟶^{∂_{C}} D^{v}(Λ^{i+1}(E)) ⟶ ⋯ (5.39)

and the cohomology theory determined by the Cartan connection. We denote the corresponding cohomology modules by H_{C}^{∗}(E) = ⨁_{i≥0} H_{C}^{i}(E). In the case of the “empty” equation, we use the notation H_{C}^{∗}(π) = ⨁_{i≥0} H_{C}^{i}(π).

Definition 5.10. Let E ⊂ J^{k}(π) be a formally integrable equation and C be the Cartan connection in the bundle π_{∞}: E^{∞} ⟶ M. Then the module H_{C}^{∗}(E) is called the C-cohomology of E.

Remark 5.5. Let us also note that the above introduced modules Λ_{h}^{q}(B) are identical to the modules Λ̄^{q}(E) of horizontal q-forms on E^{∞}, the modules Λ_{v}^{p}(B) coincide with the modules of Cartan forms C^{p}Λ(E), the differential d_{∇}^{h} is the extended horizontal de Rham differential d̄, while d_{∇}^{v} is the Cartan differential d_{C} (cf. with constructions on pp. 59–61). Thus we again obtain a complete coincidence between algebraic and geometric approaches. In particular, the ∇-spectral sequence (Definition 5.9 on page 85 (2)) is the Vinogradov C-spectral sequence (see the Section 7).

The following result contains an interpretation of the first two of Ccohomology groups.

Theorem 5.12. For any formally integrable equation E ⊂ J^{k}(π), one has the following identities:

(1) The module H_{C}^{0}(E) as a Lie algebra is isomorphic to the Lie algebra sym E of higher symmetries^{11} of the equation E.

(2) The module H_{C}^{1}(E) is the set of the equivalence classes of nontrivial vertical deformations of the equation structure (i.e., of the structural element) on E.

Proof. To prove (1), take a vertical vector field Y ∈ D^{v}(E) and an arbitrary field Z ∈ D(E). Then, due to (5.22) on page 81, one has

i_{Z}∂_{C}Y = i_{Z}[[U_{C}, Y]]^{fn} = [i_{Z}U_{C}, Y] − i_{[Z,Y]}U_{C} = [Z^{v}, Y] − [Z, Y]^{v} = [Z^{v} − Z, Y]^{v},

where Z^{v} = i_{Z}U_{C}. Hence, ∂_{C}Y = 0 if and only if [Z − Z^{v}, Y]^{v} = 0 for any Z ∈ D(E). But the last equality holds if and only if [CX, Y] = 0 for any X ∈ D(M) which means that

ker (∂_{C}: D^{v}(E) ⟶ D^{v}(Λ^{1}(E))) = sym E.

11See Definition 3.29 on page 63.

Consider the second statement now. Let U(ε) ∈ D^{v}(Λ^{1}(E)) be a deformation of the structural element satisfying the conditions [[U(ε), U(ε)]]^{fn} = 0 and U(0) = U_{C}. Then U(ε) = U_{C} + U_{1}ε + O(ε^{2}). Consequently,

[[U(ε), U(ε)]]^{fn} = [[U_{C}, U_{C}]]^{fn} + 2[[U_{C}, U_{1}]]^{fn}ε + O(ε^{2}) = 0,

from which it follows that [[U_{C}, U_{1}]]^{fn} = ∂_{C}U_{1} = 0. Hence the linear part of the deformation U(ε) determines an element of H_{C}^{1}(E) and vice versa. On the other hand, let A: E^{∞} ⟶ E^{∞} be a diffeomorphism^{12} of E^{∞}. Define the action A^{∗} of A on the elements Ω ∈ D(Λ^{∗}(E)) in such a way that the diagram

Λ^{∗}(E) ⟶^{L_{Ω}} Λ^{∗}(E)
A^{∗}↓  ↓^{A^{∗}}
Λ^{∗}(E) ⟶^{L_{Ω}} Λ^{∗}(E)

is commutative. Then, if A_{t} is a one-parameter group of diffeomorphisms, we have, obviously,

\frac{d}{dt}|_{t=0} A_{t,∗}(L_{Ω}) = \frac{d}{dt}|_{t=0} A_{t}^{∗} ∘ L_{Ω} ∘ (A_{t}^{∗})^{−1} = [L_{X}, L_{Ω}] = L_{[[X,Ω]]^{fn}}.

Hence, the infinitesimal action is given by the Frölicher–Nijenhuis bracket. Taking Ω = U_{C} and X ∈ D^{v}(E), we see that im ∂_{C} consists of infinitesimal deformations arising due to infinitesimal action of diffeomorphisms on the structural element. Such deformations are naturally called trivial.

Remark 5.6. From the general theory [14], we obtain that the module H_{C}^{2}(E) consists of obstructions to prolongation of infinitesimal deformations to formal ones. In the case under consideration, elements H_{C}^{2}(E) have another nice interpretation discussed later (see Remark 5.8 on page 94).

We shall now compute the modules H_{C}^{p}(π), p ≥ 0. To do this, recall the splitting Λ^{i}(E) = ⨁_{p+q=i} C^{p}Λ(E) ⊗ Λ̄^{q}(E) (see Subsection 5.1).

Theorem 5.13. For any p ≥ 0, one has

H_{C}^{p}(π) = F(π, π) ⊗_{F(π)} C^{p}Λ(π).

Proof. Define a filtration in D^{v}(Λ^{∗}(π)) by setting

F^{l}D^{v}(Λ^{p}(π)) = {X ∈ D^{v}(Λ^{p}(π)) | X |_{F_{l−p−1}} = 0}.

Evidently,

F^{l}D^{v}(Λ^{p}(π)) ⊂ F^{l+1}D^{v}(Λ^{p}(π)), ∂_{C}(F^{l}D^{v}(Λ^{p}(π))) ⊂ F^{l}D^{v}(Λ^{p+1}(π)).

12Since E^{∞} is, in general, infinite-dimensional, vector fields on E^{∞} do not usually possess one-parameter groups of diffeomorphisms. Thus the arguments below are of a heuristic nature.

Thus we obtain the spectral sequence associated to this filtration. To compute the term E_{0}, choose local coordinates x_{1}, …, x_{n}, u^{1}, …, u^{m} in the bundle π and consider the corresponding special coordinates u_{σ}^{j} in J^{∞}(π). In these coordinates, the structural element is represented as

U_{C} = ∑_{|σ|≥0} ∑_{j=1}^{m} (du_{σ}^{j} − ∑_{i=1}^{n} u_{σi}^{j} dx_{i}) ⊗ \frac{∂}{∂u_{σ}^{j}} ,  (5.40)

while for X = ∑_{σ,j} θ_{σ}^{j} ⊗ ∂/∂u_{σ}^{j}, θ ∈ Λ^{∗}(π), one has

∂_{C}(X) = ∑_{|σ|≥0} ∑_{j=1}^{m} ∑_{i=1}^{n} dx_{i} ∧ (θ_{σi}^{j} − D_{i}(θ_{σ}^{j})) ⊗ \frac{∂}{∂u_{σ}^{j}} .  (5.41)

Obviously, the term

E_{0}^{p,−q} = F^{p}D^{v}(Λ^{p−q}(π))/F^{p−1}D^{v}(Λ^{p−q}(π)), p ≥ 0, 0 ≤ q ≤ p,

is identified with the tensor product Λ^{p−q}(π) ⊗_{F(π)} Γ(π_{∞,q−1}^{∗}(π_{q,q−1})). These modules can be locally represented as F(π, π) ⊗ Λ^{p−q}(π)-valued homogeneous polynomials of order q, while the differential ∂_{0}^{p,−q}: E_{0}^{p,−q} ⟶ E_{0}^{p,−q+1} acts as the δ-Spencer differential (or, which is the same, as the Koszul differential; see Exercise 1.7 on page 19). Hence, all homology groups are trivial except for those at the terms E_{0}^{p,0} and one has

coker ∂_{0}^{p,0} = F(π, π) ⊗_{F(π)} C^{p}Λ(π).

Consequently, only the 0-th line survives in the term E_{1} and this line is of the form

F(π, π) ⟶^{∂_{1}^{0,0}} F(π, π) ⊗_{F(π)} C^{1}Λ(π) ⟶ ⋯

⋯ ⟶ F(π, π) ⊗_{F(π)} C^{p}Λ(π) ⟶^{∂_{1}^{p,0}} F(π, π) ⊗_{F(π)} C^{p+1}Λ(π) ⟶ ⋯

But the image of ∂_{C} contains at least one horizontal component (see equality (5.33) on page 84, where, by definition, d_{∇}^{v}−d = d_{C}−d = −d̄). Therefore, all differentials ∂_{1}^{p,0} vanish.

Let us now establish the correspondence between the last result (describing C-cohomology in terms of C^{∗}Λ(π)) and representation of H_{C}^{∗}(π) as classes of derivations F(π) ⟶ Λ^{∗}(π). To do this, for any ω = (ω^{1}, …, ω^{m}) ∈ F(π, π) ⊗_{F(π)} C^{∗}Λ(π) set

�_{ω} = ∑_{σ,j} D_{σ}(ω^{j}) ⊗ \frac{∂}{∂u_{σ}^{j}} ,  (5.42)

where D_{σ} = D_{1}^{σ_{1}} ∘ ⋯ ∘ D_{n}^{σ_{n}} for σ = (σ_{1}, … σ_{n}).

Definition 5.11. The element �_{ω} ∈ D^{v}(Λ^{∗}(π)) defined by (5.42) is called the evolutionary superderivation with the generating section ω ∈ C^{∗}Λ(π).

Proposition 5.14. The definition of �_{ω} is independent of coordinate choice.

Proof. It is easily checked that

�_{ω}(F(π)) ⊂ Λ_{v}^{∗}(π), �_{ω} ∈ ker ∂_{C}.

But derivations possessing these properties are uniquely determined by their restriction to F_{0}(π) which coincides with the action of the derivation ω : F_{0}(π) ⟶ C^{∗}Λ(π). Let us prove this fact.

Set X = �_{ω} and recall that the derivation X is uniquely determined by the corresponding Lie derivative L_{X} : Λ^{∗}(π) ⟶ Λ^{∗}(π). Further, since L_{X}dθ = (−1)^{X}d(L_{X}θ) (see (5.16) on page 80) for any θ ∈ Λ^{∗}(π), the derivation L_{X} is determined by its restriction to Λ^{0}(π) = F(π).

Now, from the identity ∂_{C}X = 0 it follows that

0 = [[U_{C}, X]]^{fn}(f) = L_{U_{C}}(L_{X}(f)) − (−1)^{X}L_{X}(L_{U_{C}}(f)), f ∈ F(E). (5.43)

Let now X be such that L_{X} |_{F_{0}(π)} = 0 and assume that L_{X} |_{F_{r}(π)} = 0 for some r > 0. Then taking f = u_{σ}^{j}, |σ| = r, and using (5.43) we obtain

L_{X} (du_{σ}^{j} − ∑_{i=1}^{n} u_{σi}^{j}dx_{i}) = L_{X}d_{C}u_{σ}^{j} = (−1)^{X}d_{C}(L_{X}(u_{σ}^{j})) = 0.

In other words,

L_{X} (∑_{i=1}^{n} u_{σi}^{j}dx_{i}) = ∑_{i=1}^{n} L_{X}(u_{σi}^{j}dx_{i})) = ∑_{i=1}^{n} L_{X}(du_{σ}^{j})

= (−1)^{X} ∑_{i=1}^{n} d(L_{X}u_{σ}^{j}) = 0.

Hence, L_{X}(u_{σ}^{j}) = 0 and thus L_{X} |_{F_{r+1}(π)} = 0.

From this result and from Corollary 5.10 on page 84, it follows that if two evolutionary superderivations �_{ω}, �_{θ} are given, the elements

(i) [[�_{ω} , �_{θ} ]]^{fn}, (ii) i_{�_{ω}} (�_{θ})

are evolutionary superderivations as well.

In the first case, the corresponding generating section is called the Jacobi superbracket of elements ω = (ω^{1}, …, ω^{m}) and θ = (θ^{1}, …, θ^{m}) and is denoted by {ω, θ}. The components of this bracket are expressed by the formula

{ω, θ}^{j} = L_{�_{ω}} (θ^{j}) − (−1)^{ωθ}L_{�_{θ}} (ω^{j}), j = 1, …, m.  (5.44)

Obviously, the module F(π, π) ⊗_{F(π)} C^{∗}Λ(π) is a graded Lie algebra with respect to the Jacobi superbracket. The restriction of {·, ·} to F(π, π) ⊗ C^{0}Λ(π) = F(π, π) coincides with the higher Jacobi bracket (see Definition 3.31 on page 65).

In the case (ii), the generating section is i_{�_{ω}} (θ). Note now that any element ρ ∈ C^{1}Λ(π) is of the form ρ = ∑_{σ,α} a_{σ,α}ω_{σ}^{α}, where, as before, ω_{σ}^{α} = d_{C}u_{σ}^{α} = du_{σ}^{α} − ∑_{i=1}^{n} u_{σi}^{α}dx_{i} are the Cartan forms on J^{∞}(π). Hence, if θ ∈ F(π, π) ⊗_{F(π)} C^{1}Λ(π) and θ^{j} = ∑_{σ,α} a_{σ,α}^{j}ω_{σ}^{α}, then

(i_{�_{ω}} (θ))^{j} = ∑_{σ,α} a_{σ,α}^{j}D_{σ}(ω^{α}).  (5.45)

In particular, we see that (5.45) establishes an isomorphism between the modules F(π, π) ⊗_{F(π)} C^{∗}Λ(π) and CDiff(π, π) and defines the action of Cdifferential operators on elements of C^{∗}Λ(π). This is a really well-defined action because of the fact that i_{CX}ω = 0 for any X ∈ D(M) and ω ∈ C^{∗}Λ(π).

Consider now a formally integrable differential equation E ⊂ J^{k}(π) and assume that it is determined by a differential operator Δ ∈ F(π, ξ). Denote, as in Section 3, by ℓ_{E} the restriction of the operator of universal linearization ℓ_{Δ} to E^{∞}. Let ℓ_{E}^{[p]} be the extension of ℓ_{E} to F(π, π) ⊗_{F(π)} C^{p}Λ(E) which is well defined due to what has been said above. Then the module H_{C}^{p,0}(E) is identified with the set of evolution superderivations �_{ω} whose generating sections ω ∈ F(π, π) ⊗_{F(π)} C^{p}Λ(E) satisfy the equation

ℓ_{E}^{[p]}(ω) = 0  (5.46)

If, in addition, E satisfies the assumptions of the two-line theorem, then H_{C}^{p,1}(E) is identified with the cokernel of ℓ_{E}^{[p−1]} and thus

H_{C}^{i}(E) = ker ℓ_{E}^{[i]} ⊕ coker ℓ_{E}^{[i−1]}

in this case. These two statements will be proved in Subsection 6.4.

As it was noted in Remark 5.4 on page 85, H_{C}^{1}(E) is an associative algebra with respect to contraction and is represented in the algebra of endomorphisms of H_{C}^{0}(E). It is easily seen that the action of the H_{C}^{0,1}(E) is trivial while H_{C}^{1,0}(E) acts on H_{C}^{0}(E) = sym E as C-differential operators (see above).

Definition 5.12. Elements of the module H_{C}^{1,0}(E) are called recursion operators for symmetries of the equation E.

We use the notation R(E) for the algebra of recursion operators.

Remark 5.7. The algebra R(E) is always nonempty, since it contains the structural element U_{E} which is the unit of this algebra. “Usually” this is the only solution of (5.46) for p = 1 (see Example 5.1 below). This fact apparently contradicts practical experience (cf. with well-known recursion operators for the KdV and other integrable systems [43]). The reason is that these operators contain nonlocal terms like D^{−1} or of a more complicated form. An adequate framework to deal with such constructions will be described in the next subsection.

Example 5.1. Let

u_{t} = uu_{x} + u_{xx}  (5.47)

be the Burgers equation. For internal coordinates on E^{∞} we choose the functions x, t, u = u_{0}, …, u_{k}, …, where u_{k} corresponds to the partial derivative ∂^{k}u/∂x^{k}.

We shall prove here that the only solution of the equation ℓ_{E}^{[1]}(ω) = 0 for (5.47) is ω = αω_{0}, α = const, where

ω_{k} = d_{C}u_{k} = du_{k} − u_{k+1}dx − D_{x}^{k}(uu_{1} + u_{2})dt.

Let ω = ϕ^{0}ω_{0} + ⋯ + ϕ^{r}ω_{r}. Then the equation (5.46) on the page before for p = 1 transforms to

u_{0}D_{x}(ϕ^{0}) + D_{x}^{2}(ϕ^{0}) = D_{t}(ϕ^{0}) + ∑_{j=1}^{r} u_{j+1}ϕ^{j},

u_{0}D_{x}(ϕ^{1}) + D_{x}^{2}(ϕ^{1}) + 2D_{x}(ϕ^{0}) = D_{t}(ϕ^{1}) + ∑_{j=2}^{r}(j + 1)u_{j}ϕ^{j},

…  (5.48)

u_{0}D_{x}(ϕ^{i}) + D_{x}^{2}(ϕ^{i}) + 2D_{x}(ϕ^{0}) = D_{t}(ϕ^{i}) + ∑_{j=i+1}^{r} \binom{j + 1}{i}u_{j−i+1}ϕ^{j},

…

u_{0}D_{x}(ϕ^{r}) + D_{x}^{2}(ϕ^{r}) + 2D_{x}(ϕ^{r−1}) = D_{t}(ϕ^{r}) + ru_{1}ϕ^{r},

D_{x}(ϕ^{r}) = 0.

To prove the result, we apply the scheme used in [64] to describe the symmetry algebra of the Burgers equation.

Denote by K_{r} the set of solutions of (5.48). Then a direct computation shows that

K^{1} = {αω_{0} | α ∈ ℝ}  (5.49)

and that any element ω ∈ K_{r}, r > 1, is of the form

ω = α_{r} + (\frac{r}{2}u_{0}α_{r} + \frac{1}{2}xα_{r}^{(1)} + α_{r−1}) + Ω[r − 2],  (5.50)

where α_{r} = α_{r}(t), α_{r−1} = α_{r−1}(t), α^{(i)} denotes the derivative d^{i}α/dt^{i}, and Ω[s] is an arbitrary linear combination of ω_{0}, …, ω_{s} with coefficients in F(E).

Note now that for any evolution equation the embedding

[[sym E, ker ℓ_{E}^{[1]}]]^{fn} ⊂ ker ℓ_{E}^{[1]}

is valid. Consequently, if ψ ∈ sym E and ω ∈ ker ℓ_{E}^{[1]}, then {ψ, ω} ∈ ker ℓ_{E}^{[1]}.

Since the function u_{1} is a symmetry of the Burgers equation (translation along x), one has

\{u_{1}, ω\} = (∑_{k} u_{k+1} \frac{∂}{∂u_{k}} ) ω − D_{x}ω = −\frac{∂}{∂x}ω.

Hence, if ω ∈ K_{r}, then from (5.50) we obtain that

ad_{u_{1}}^{(r−1)}(ω) = α_{r}^{(r−1)}ω_{1} + Ω[0] ∈ K_{1},

where ad_{ψ} = {ψ, ·}. Taking into account equation (5.49), we get that α_{r}^{r−1} = 0, or

α_{r} = a_{0} + a_{1}t + ⋯ + a_{r−2}t^{r−2}, a_{i} ∈ ℝ.  (5.51)

We shall use now the fact that the element Φ = t^{2}u_{2}+(t^{2}u_{0}+tx)u_{1}+tu_{0}+1 is a symmetry of the Burgers equation (see [64]). Then, since the action of symmetries is permutable with the Cartan differential d_{C}, we have

{Φ, ϕ^{s}ω_{s}} = �_{Φ}(ϕ^{s}ω_{s}) − �_{ϕ^{s}ω_{s}}(Φ) = �_{Φ}(ϕ^{s})ω_{s} + ϕ^{s}�_{Φ}(ω_{s}) − �_{ϕ^{s}ω_{s}}(Φ).

But

�_{Φ}(ω_{s}) = �_{Φ}d_{C}(u_{s}) = d_{C}�_{Φ}(u_{s}) = d_{C}D_{x}^{s}(Φ)

= d_{C}(t^{2}u_{s+2} + (t^{2}u_{0} + tx)u_{s+1} + (s + 1)(t^{2}u_{1} + t)u_{s}) + Ω[s − 1].

On the other hand,

�_{ϕ^{s}ω_{s}}(Φ) = t^{2}ϕ^{s}ω_{s+2} + (2t^{2}D_{x}^{2}(ϕ^{s}) + (t^{2}u_{0} + tx)ϕ^{s})ω_{s+1}

+ (t^{2}D_{x}^{2}(ϕ^{s}) + (t^{2}u_{0} + tx)D_{x}(ϕ^{s}) + (t^{2}u_{1} + t)ϕ^{s})ω_{s}.

Thus, we finally obtain

\begin{aligned}\{Φ, ϕ^{s}ω_{s}\} = \{Φ, ϕ^{s}\}ω_{s} + (s + 1)(t^{2}u_{1} & + t)ω_{s} \\ & − 2t^{2}D_{x}(ϕ^{s})ω_{s+1} + Ω[s − 1]. \end{aligned} (5.52)

Applying (5.52) to (5.50), we get

ad_{Φ}(ω) = (rtα_{r} − t^{2}α_{r}^{(1)})ω_{r} + Ω[r − 1].  (5.53)

Let now ω ∈ K_{r} and assume that ω has a nontrivial coefficient α_{r} of the form (5.51), and a_{i} be the first nontrivial coefficient in α_{r}. Then, by representation (5.53), we have

ad_{Φ}^{r−i}(ω) = α′_{r}ω_{r} + Ω[r − 1] ∈ K_{r},

where α′_{r} is a polynomial of degree r − 1. This contradicts to (5.51) and thus concludes the proof.

Remark 5.8. Let φ ∈ sym E be a symmetry and R ∈ R(E) be a recursion operator. Then we obtain a sequence of symmetries φ_{0} = φ, φ_{1} = R(φ), …, φ_{n} = R^{n}(φ), . . .. Using identity (5.22) on page 81, one can compute the commutators [φ_{m}, φ_{n}] in terms of [[φ, R]]^{fn} ∈ H_{C}^{1,0}(E) and [[R, R]]^{fn} ∈ H_{C}^{2,0}(E). In particular, it can be shown that when both [[φ, R]]^{fn} and [[R, R]]^{fn} vanish, all symmetries φ_{n} mutually commute (see [27]).

For example, if E is an evolution equation, H_{C}^{p,0}(E) = 0 for all p ≥ 2 (see Theorem 6.8 on page 110). Hence, if φ is a symmetry and R is a φinvariant recursion operator (i.e., such that [[φ, R]]^{fn} = 0), then R generates a commutative sequence of symmetries. This is exactly the case for the KdV and other integrable evolution equations.

5.4. Passing to nonlocalities. Let us now introduce nonlocal variables into the above described picture. Namely, let E be an equation and φ : N ⟶ E^{∞} be a covering over its infinite prolongation. Then, due to Proposition 4.1 on page 69, the triad (F(N), C^{∞}(M), (π_{∞} ∘ φ)^{∗}) is an algebra with the flat connection C^{φ}. Hence, we can apply the whole machinery of Subsections 5.1–5.3 to this situation. To stress the fact that we are working over the covering φ, we shall add the symbol φ to all notations introduced in these subsections. Denote by U_{C}^{φ} the connection form of the connection C^{φ} (the structural element of the covering φ).

In particular, on N we have the C^{φ}-differential ∂_{C}^{φ} = [[U_{C}^{φ}, ·]]^{fn} : D^{v}(Λ^{i}(N)) → D^{v}(Λ^{i+1}(N)), whose 0-cohomology H_{C}^{0}(E, φ) coincides with the Lie algebra sym_{φ} E of nonlocal φ-symmetries, while the module H_{C}^{1,0}(E, φ) identifies with recursion operators acting on these symmetries and is denoted by R(E, φ). We also have the horizontal and the Cartan differential d̄^{φ} and d_{C}^{φ} on N and the splitting Λ^{i}(N) = ⨁_{p+q=i} C^{p}Λ^{p}(N) ⊗ Λ̄^{q}(N).

Choose a trivialization of the bundle φ : N ⟶ E^{∞} and nonlocal coordinates w^{1}, w^{2}, … in the fiber. Then any derivation X ∈ D^{v}(Λ^{i}(N)) splits to the sum X = X_{E} +X^{v}, where X_{E}(w^{j}) = 0 and X^{v} is a φ-vertical derivation.

Lemma 5.15. Let φ : E^{∞} × ℝ^{N} ⟶ E^{∞}, N ≤ ∞, be a covering. Then H_{C}^{p,0}(E, φ) = ker ∂_{C}^{φ} |_{C^{p}Λ(N)} . Thus H_{C}^{p,0}(E, φ) consists of derivations Ω : F(N) ⟶ C^{p}Λ(N) such that

[[U_{C}^{φ}, Ω]]_{E}^{fn} = 0, ([[U_{C}^{φ}, Ω]]^{fn})^{v} = 0.  (5.54)

Proof. In fact, due to equality (5.33) on page 84, any element lying in the image of ∂_{C}^{φ} contains at least one horizontal component, i.e.,

∂_{C}^{φ}(D^{v}(C^{p}Λ(N))) ⊂ D^{v}(C^{p}Λ(N) ⊗ Λ̄^{1}(N)).

Thus, equations (5.54) should hold.

We call the first equation in (5.54) on the facing page the shadow equation while the second one is called the relation equation. This is explained by the following result (cf. with Theorem 4.7 on page 72).

Proposition 5.16. Let E be an evolution equation of the form

u_{t} = f(x, t, u, …, \frac{∂^{k}u}{∂u^{k}})

and φ : N = E^{∞} × ℝ^{N} ⟶ E^{∞} be a covering given by the vector fields^{13}

D̃_{x} = D_{x} + X, D̃_{t} = D_{t} + T,

where [D̃_{x}, D̃_{t}] = 0 and

X = ∑_{s} X^{s} \frac{∂}{∂w^{s}} , T = ∑_{s} T^{s} \frac{∂}{∂w^{s}} ,

w^{1}, …, w^{s}, … being nonlocal variables in φ. Then the group H_{C}^{p,0}(E, φ) consists of elements

Ψ = ∑_{i} Ψ_{i} ⊗ \frac{∂}{∂u_{i}} + ∑_{s} ψ^{s} \frac{∂}{∂w^{s}} ∈ D^{v}(C^{p}Λ(N))

such that Ψ_{i} = D̃_{x}^{i}Ψ_{0} and

ℓ̃_{E}^{[p]}(Ψ_{0}) = 0,  (5.55)
∑_{α} \frac{∂X^{s}}{∂u_{α}} D̃_{x}^{α}(Ψ_{0}) + ∑_{β} \frac{∂X^{s}}{∂w^{β}} ψ^{β} = D̃_{x}(ψ^{s}),  (5.56)

∑_{α} \frac{∂T^{s}}{∂u_{α}} D̃_{x}^{α}(Ψ_{0}) + ∑_{β} \frac{∂T^{s}}{∂w^{β}} ψ^{β} = D̃_{t}(ψ^{s}),  (5.57)

s = 1, 2, …, where ℓ̃_{E}^{[p]} is the natural extension of the operator ℓ_{E}^{[p]} to N.

Proof. Consider the Cartan forms

ω_{i} = du_{i} − u_{i+1}dx − D_{x}^{i}(f) dt, θ^{s} = dw^{s} − X^{s}dx − T^{s}dt

on N. Then the derivation

U_{C}^{φ} = ∑_{i} ω_{i} ⊗ \frac{∂}{∂u_{i}} + ∑_{s} θ^{s} ⊗ \frac{∂}{∂w^{s}}

13To simplify the notations of Section 4, we denote the lifting of a C-differential operator Δ to N by Δ̃.

is the structural element of the covering φ. Then, using representation (5.18) on page 80, we obtain

∂_{C}^{φ}Ψ = dx ∧ ∑_{i} (Ψ_{i+1} − D̃_{x}(Ψ_{i})) ⊗ \frac{∂}{∂u_{i}}

+ dt ∧ ∑_{i} ( ∑_{α} \frac{∂(D_{x}^{i}f)}{∂u_{α}}Ψ_{α} − D̃_{t}Ψ_{i}) ⊗ \frac{∂}{∂u_{i}}

+ dx ∧ ∑_{s} ( ∑_{α} \frac{∂X^{s}}{∂u_{α}} Ψ_{α} + ∑_{β} \frac{∂X^{s}}{∂w^{β}} ψ^{β} − D̃_{x}(ψ^{s})) ⊗ \frac{∂}{∂w^{s}}

+ dt ∧ ∑_{s} ( ∑_{α} \frac{∂T^{s}}{∂u_{α}} Ψ_{α} + ∑_{β} \frac{∂T^{s}}{∂w^{β}} ψ^{β} − D̃_{t}(ψ^{s})) ⊗ \frac{∂}{∂w^{s}} ,

which gives the needed result.

Note that relations Ψ_{i} = D̃_{x}^{i}(Ψ_{0}) together with equation (5.55) are equivalent to the shadow equations. In the case p = 1, we call the solutions of equation (5.55) the shadows of recursion operators in the covering φ. Equations (5.56) and (5.57) on the page before are exactly the relation equations on the case under consideration.

Exercise 5.1. Generalize the above result to general equations using the proof similar to that of Theorem 4.7 on page 72.

Thus, any element of the group H_{C}^{1,0}(E, φ) is of the form

Ψ = ∑_{i} D̃_{x}^{i}(ψ) ⊗ \frac{∂}{∂u_{i}} + ∑_{s} ψ^{s} ⊗ \frac{∂}{∂w^{s}} ,  (5.58)

where the forms ψ = Ψ_{0}, ψ^{s} ∈ C^{1}Λ(N) satisfy the system of equations (5.55)–(5.57).

As a direct consequence of the above said, we obtain the following

Corollary 5.17. Let Ψ be a derivation of the form (5.58) with ψ, ψ^{s} ∈ C^{p}Λ(N). Then ψ is a solution of equation (5.55) on the preceding page in the covering φ if and only if ∂_{C}^{φ}(Ψ) is a φ-vertical derivation.

We can now formulate the main result of this subsection.

Theorem 5.18. Let φ : N ⟶ E^{∞} be a covering, S ∈ sym_{φ} E be a φsymmetry, and ψ ∈ C^{1}Λ(N) be a shadow of a recursion operator in the covering φ. Then ψ′ = i_{S}ψ is a shadow of a symmetry in φ, i.e., ℓ̃_{E}(ψ′) = 0.

Proof. In fact, let Ψ be a derivation of the form (5.58). Then, due to identity (5.32) on page 84, one has

∂_{C}^{φ}(i_{S}Ψ) = i_{∂_{C}^{φ}S} − i_{S}(∂_{C}^{φ}Ψ) = −i_{S}(∂_{C}^{φ}Ψ),

since S is a symmetry. But, by Corollary 5.17 on the facing page, ∂_{C}^{φ}Ψ is a φ-vertical derivation and consequently ∂_{C}^{φ}(i_{S}Ψ) = −i_{S}(∂_{C}^{φ}Ψ) is φ-vertical as well. Hence, i_{S}Ψ is a φ-shadow by the same corollary.

Using the last result together with Theorem 4.11 on page 76, we can describe the process of generating a series of symmetries by shadows of recursion operators. Namely, let ψ be a symmetry and ω ∈ C^{1}Λ(N) be a shadow of a recursion operator in a covering φ : N ⟶ E^{∞}. In particular, ψ is a φ-shadow. Then, by Theorem 4.9 on page 75, there exists a covering φ_{ψ} : N_{ψ} ⟶ N ⟶^{φ} E^{∞} where �_{ψ} can be lifted to as a φ_{ψ}-symmetry. Obviously, ω still remains a shadow in this new covering. Therefore, we can act by ω on ψ and obtain a shadow ψ_{1} of a new symmetry on N_{ψ}. By Theorem 4.11 on page 76, there exists a covering, where both ψ and ψ_{1} are realized as nonlocal symmetries. Thus we can continue the procedure applying ω to ψ_{1} and eventually arrive to a covering in which the whole series {ψ_{k}} is realized.

Example 5.2. Let u_{t} = uu_{x} + u_{xx} be the Burgers equation. Consider the one-dimensional covering φ : E^{∞} × ℝ ⟶ E^{∞} with the nonlocal variable w and defined by the vector fields

D_{x}^{φ} = D_{x} + u_{0} \frac{∂}{∂w}, D_{t}^{φ} = D_{t} + (\frac{u_{0}^{2}}{2} + u_{1}) \frac{∂}{∂w}.

Then it easily checked that the form

ω = ω_{1} + \frac{1}{2}ω_{0} + \frac{1}{2}θ,

where ω_{0} and ω_{1} are the Cartan forms d_{C}u_{0} and d_{C}u_{1} respectively and θ = dw − u_{0}dx − (u_{0}^{2}/2 + u_{1})dt, is a solution of the equation ℓ̃_{E}^{[1]}ω = 0. If �_{ψ} is a symmetry of the Burgers equation, the corresponding action of ω on ψ is

D_{x}ψ + \frac{1}{2}ψ + \frac{1}{2}D_{x}^{−1}ψ

and thus coincides with the well-known recursion operator for this equation, see [43].

Exercise 5.2. Let u_{t} = uu_{x} + u_{xxx} be the KdV equation. Consider the onedimensional covering φ : E^{∞} × ℝ ⟶ E^{∞} with the nonlocal variable w and defined by the vector fields

D_{x}^{φ} = D_{x} + u_{0} \frac{∂}{∂w}, D_{t}^{φ} = D_{t} + (\frac{u_{0}^{2}}{2} + u_{2}) \frac{∂}{∂w}.

Solve the equation ℓ̃_{E}^{[1]}ω = 0 in this covering and find the corresponding recursion operator.

Remark 5.9. Recursion operators can be understood as supersymmetries (cf. Subsection 7.9 on page 130) of a certain superequation naturally related to the initial one. To such symmetries and equations one can apply nonlocal theory of Section 4 and prove the corresponding reconstruction theorems, see [28, 30].

6. Horizontal cohomology

In this section we discuss the horizontal cohomology of differential equations, i.e., the cohomology of the horizontal de Rham complex (see Definition 3.27 on page 59). This cohomology has many physically relevant applications. To demonstrate this, let us start with the notion of a conserved current. Consider a differential equation E. A conserved current is a vector-function J = (J_{1}, …, J_{n}), where J_{k} ∈ F(E), which is conserved modulo the equation, i.e., that satisfies the equation

∑_{k=1}^{n} D_{k}(J_{k}) = 0,  (6.1)

where D_{k} are restrictions of total derivatives to E^{∞}. For example, take the nonlinear Schrödinger equation^{14}

iψ_{t} = Δψ + |ψ|^{2}ψ,  Δ = ∑_{j=1}^{n−1} \frac{∂^{2}}{∂x_{j}^{2}} .  (6.2)

Then it is straightforwardly verified, that the vector-function

J = (|ψ|^{2}, i(ψ̄ψ_{x_{1}} − ψψ̄_{x_{1}}), …, i(ψ̄ψ_{x_{n−1}} − ψψ̄_{x_{n−1}}))

is a conserved current, i.e., that

D_{t}(|ψ|^{2}) + ∑_{k=1}^{n−1} D_{k}(i(ψ̄ψ_{x_{k}} − ψψ̄_{x_{k}}))

vanishes by virtue of equation (6.2).

A conserved current is called trivial, if it has the form

J_{k} = ∑_{l=1}^{n} D_{l}(L_{kl})  (6.3)

for some skew-symmetric matrix, ∥L_{kl}∥, L_{kl} = −L_{lk}, L_{kl} ∈ F(E). The name “trivial currents” means that they are trivially conserved regardless to the equation under consideration. Two conserved currents are said to be equivalent if they differ by a trivial one. Conservation laws are defined to be the equivalent classes of conserved currents.

Let us assign the horizontal (n−1)-form ω_{J} = ∑_{k=1}^{n}(−1)^{k−1}J_{k} dx_{1} ∧⋯∧ d̂x_{k} ∧ ⋯ ∧ dx_{n} to each conserved current J = (J_{1}, …, J_{n}). Then equations (6.1) and (6.3) can be rewritten as d̄ω_{J} = 0 and ω_{J} = d̄η respectively, where η = ∑_{k > l}(−1)^{k+l}L_{kl} dx_{1} ∧ ⋯ ∧ d̂x_{l} ∧ ⋯ ∧ d̂x_{k} ∧ ⋯ ∧ dx_{n}. Thus, we see that the horizontal cohomology group in degree n − 1 of the equation E consists of conservation laws of E.

14Here ψ is a complex function and (6.2) is to be understood as a system of two equations.

In physical applications one also encounters the horizontal cohomology in degree less than n − 1. For instance, the Maxwell equations read

d̄(∗F) = 0,

where F is the electromagnetic field strength tensor and ∗ is the Hodge star operator. Clearly ∗F is not exact. Another reason to consider the low-dimensional horizontal cohomology is that it appears as an auxiliary cohomology in calculation of the BRST cohomology [5]. Recently, by means of horizontal cohomology the problem of consistent deformations and of candidate anomalies has been completely solved in cases of Yang-Mills gauge theories and of gravity [6, 4].

The horizontal cohomology plays a central role in the Lagrangian formalism as well. Really, it is easy to see that the horizontal cohomology group in degree n is exactly the space of actions of variational problems constrained by equation E.

For computing the horizontal cohomology there is a general method based on the Vinogradov C-spectral sequence. It can be outlined as follows. The horizontal cohomology is the term E_{1}^{0,•} of the Vinogradov C-spectral sequence and thereby related to the terms E_{1}^{p,•}, p > 0. For each p, such a term is also a horizontal cohomology but with some nontrivial coefficients. The crucial observation is that the corresponding modules of coefficients are supplied with filtrations such that the differentials of the associated graded complexes are linear over the functions. Hence, the cohomology can be computed algebraically. A detailed description of these techniques is our main concern in this and the next sections.

6.1. C-modules on differential equations. Let us begin with the definition of C-modules, which are left differential modules (see Definition 1.7 on page 16) in C-differential calculus and serve as the modules of coefficients for horizontal de Rham complexes.

Proposition 6.1. The following three definitions of a C-module are equivalent:

(1) An F-module Q is called a C-module, if Q is endowed with a left module structure over the ring CDiff(F, F), i.e., for any scalar C-dif-ferential operator Δ ∈ CDiff_{k}(F, F) there exists an operator Δ_{Q} ∈ CDiff_{k}(Q, Q), with

(1) (∑_{i} f_{i}Δ_{i})_{Q} = ∑_{i} f_{i}(Δ_{i})_{Q}, f_{i} ∈ F,

(2) (id_{F})_{Q} = id_{Q},

(3) (Δ_{1} ∘ Δ_{2})_{Q} = (Δ_{1})_{Q} ∘ (Δ_{2})_{Q}.

(2) A C-module is a module equipped with a flat horizontal connection, i.e., with an action on Q of the module CD = CD(E), X ↦ ∇_{X}, which is F-linear:

∇_{fX+gY} = f∇_{X} + g∇_{Y} , f, g ∈ F, X, Y ∈ CD,

satisfies the Leibniz rule:

∇_{X}(fq) = X(f)q + f∇_{X}(q), q ∈ Q, X ∈ CD, f ∈ F,

and is a Lie algebra homomorphism:

[∇_{X}, ∇_{Y}] = ∇_{[X,Y]}.

(3) A C-module is the module of sections of a linear covering, i.e., Q is the module of sections of a vector bundle τ: W → E^{∞}, Q = Γ(τ), equipped with a completely integrable n-dimensional linear distribution (see Definition 4.3 on page 69) on W which is projected onto the Cartan distribution on E^{∞}.

The proof is elementary.

Exercise 6.1. Show that

(1) in coordinates, the operator (D_{i})_{Q} = ∥Δ_{j}^{k}∥ is a matrix operator of the form

Δ_{j}^{k} = D_{i}δ_{j}^{k} + Γ_{ij}^{k}, Γ_{ij}^{k} ∈ F,

where δ_{j}^{k} is the Kronecker symbol;

(2) the coordinate description of the corresponding flat horizontal connection looks as

∇_{D_{i}}(s_{j}) = ∑_{k} Γ_{ij}^{k}s_{k}

where s_{j} are basis elements of Q;

(3) the corresponding linear covering has the form

D̃_{i} = D_{i} + ∑_{j,k} Γ_{ij}^{k}w^{j} \frac{∂}{∂w^{k}} ,

where w^{i} are fiber coordinates on W.

Here are basic examples of C-modules.

Example 6.1. The simplest example of a C-module is Q = F with the usual action of C-differential operators.

Example 6.2. The module of vertical vector fields Q = D^{v} = D^{v}(E) with the connection

∇_{X}(Y) = [X, Y]^{v}, X ∈ CD, Y ∈ D^{v},

where Z^{v} = U_{C}(Z), is a C-module.

Example 6.3. Next example is the modules of Cartan forms Q = C^{k}Λ = C^{k}Λ(E). A vector field X ∈ CD acts on C^{k}Λ as the Lie derivative L_{X}. It is easily seen that in coordinates we have

(D_{i})_{C^{k}Λ}(ω_{σ}^{j}) = ω_{σi}^{j}.

Example 6.4. The infinite jet module Q = J̄^{∞}(P) of an F-module P is a C-module via

Δ_{J̄^{∞}(P)}(fj̄_{∞}(p)) = Δ(f)j̄_{∞}(p),

where Δ ∈ CDiff(F, F), f ∈ F, p ∈ P.

Example 6.5. Let us dualize the previous example. It is clear that for any F-module P the module Q = CDiff(P, F) is a C-module. The action of horizontal operators is the composition:

Δ_{Q}(∇) = Δ ∘ ∇,

where Δ ∈ CDiff(F, F), ∇ ∈ Q = CDiff(P, F).

Example 6.6. More generally, let Δ: P → P_{1} be a C-differential operator and ψ_{∞}^{Δ} : J̄^{∞}(P) → J̄^{∞}(P_{1}) be the corresponding prolongation of Δ. Obviously, ψ_{∞}^{Δ} is a morphism of C-modules, i.e., a homomorphism over the ring CDiff(F, F), so that ker ψ_{∞}^{Δ} and coker ψ_{∞}^{Δ} are C-modules.

On the other hand, the operator Δ gives rise to the morphism of C-mod-ules CDiff(P_{1}, F) → CDiff(P, F), ∇ ↦ ∇∘Δ. Thus the kernel and cokernel of this map are C-modules as well.

Example 6.7. Given two C-modules Q_{1} and Q_{2}, we can define C-module structures on Q_{1} ⊗_{F} Q_{2} and Hom_{F}(Q_{1}, Q_{2}) by

∇_{X}(q_{1} ⊗ q_{2}) = ∇_{X}(q_{1}) ⊗ q_{2} + q_{1} ⊗ ∇_{X}(q_{2}),
∇_{X}(f)(q_{1}) = ∇_{X}(f(q_{1})) − f(∇_{X}(q_{1})),

where X ∈ CD, q_{1} ∈ Q_{1}, q_{2} ∈ Q_{2}, f ∈ Hom_{F}(Q_{1}, Q_{2}).

For instance, one has C-module structures on Q = J̄^{∞}(P) ⊗_{F} C^{k}Λ and Q = CDiff(P, C^{k}Λ) for any F-module P.

Example 6.8. Let 𝔤 be a Lie algebra and ρ: 𝔤 → 𝔤𝔩(W) a linear representation of 𝔤. Each 𝔤-valued horizontal form ω ∈ Λ̄^{1}(E)⊗_{ℝ} 𝔤 that satisfies the horizontal Maurer–Cartan condition d̄ω + \frac{1}{2}[ω, ω] = 0 defines on the module Q of sections of the trivial vector bundle E^{∞} × W → E^{∞} the following C-module structure:

∇_{X}(q)_{a} = X(q)_{a} + ρ(ω(X))(q_{a}),

where X ∈ CD, q ∈ Q, a ∈ E^{∞}, and X(q) means the component-wise action.

Exercise 6.2. Check that Q is indeed a C-module.

Such C-modules are called zero-curvature representations over E^{∞}. Take the example of the KdV equation (in the form u_{t} = uu_{x} + u_{xxx}) and 𝔤 = 𝔰𝔩_{2}(ℝ). Then there exists a one-parameter family of Maurer–Cartan forms ω(λ) = A_{1}(λ) d̄x + A_{2}(λ) d̄t, λ being a parameter:

A_{1}(λ) = (\begin{matrix}0 & −(λ + u) \\ \frac{1}{6} & 0 \end{matrix})

and

A_{2}(λ) = (\begin{matrix}−\frac{1}{6}u_{x} & −u_{xx} − \frac{1}{3}u^{2} + \frac{1}{3}λu + \frac{2}{3}λ^{2} \\ \frac{1}{18}u − \frac{1}{9}λ & \frac{1}{6}u_{x} \end{matrix}) .

This is the zero-curvature representation used in the inverse scattering method.

Remark 6.1. In parallel with left C-modules one can consider right C-mod-ules, i.e., right modules over the ring CDiff(F, F). There is a natural way to pass from left C-modules to right ones and back. Namely, for any left module Q set

B(Q) = Q ⊗_{F} Λ̄^{n}(E),

with the right action of CDiff(F, F) on B(Q) given by

(q ⊗ ω)f = fq ⊗ ω = q ⊗ fω, f ∈ F,
(q ⊗ ω)X = −∇_{X}(q) ⊗ ω − q ⊗ L_{X}ω, X ∈ CD.

One can easily verify that B determines an equivalence between the categories of left C-modules and right C-modules.

By definition of a C-module, for a scalar C-differential operator Δ: F → F there exists the extension Δ_{Q}: Q → Q of Δ to the C-module Q. Similarly to Lemma 1.16 on page 16 one has more: for any C-differential operator Δ: P → S there exists the extension Δ_{Q}: P ⊗_{F} Q → S ⊗_{F} Q.

Proposition 6.2. Let P, S be F-modules. Then there exists a unique mapping

CDiff_{k}(P, S) → CDiff_{k}(P ⊗_{F} Q, S ⊗_{F} Q), Δ ↦ Δ_{Q},

such that the following conditions hold:

(1) if P = S = F then the mapping is given by the C-module structure on Q,

(2) (∑_{i} f_{i}Δ_{i})_{Q} = ∑_{i} f_{i}(Δ_{i})_{Q}, f_{i} ∈ F,

(3) if Δ ∈ CDiff_{0}(P, S) = Hom_{F}(P, S) then Δ_{Q} = Δ ⊗_{F} id_{Q},

(4) if R is another F-module and Δ_{1} : P → S, Δ_{2} : S → R are C-differ-ential operators, then (Δ_{2} ∘ Δ_{1})_{Q} = (Δ_{2})_{Q} ∘ (Δ_{1})_{Q}.

Proof. The uniqueness is obvious. To prove the existence consider the family of operators Δ(p, s^{∗}): F → F, p ∈ P, s^{∗} ∈ S^{∗} = Hom_{F}(S, F), Δ(p, s^{∗})(f) = s^{∗}(Δ(fp)), f ∈ F. Clearly, the operator Δ is defined by the family Δ(p, s^{∗}). The following statement is also obvious.

Exercise 6.3. For the family of operators Δ[p, s^{∗}] ∈ CDiff_{k}(F, F), p ∈ P, s^{∗} ∈ S^{∗}, we can find an operator Δ ∈ CDiff_{k}(P, S) such that Δ[p, s^{∗}] = Δ(p, s^{∗}), if and only if

Δ[p, ∑_{i} f_{i}s_{i}^{∗}] = ∑_{i} f_{i}Δ[p, s_{i}^{∗}],
Δ[∑_{i} f_{i}p_{i}, s^{∗}] = ∑_{i} Δ[p_{i}, s^{∗}]f_{i}.

In view of this exercise, the family of operators

Δ_{Q}[p ⊗ q, s^{∗} ⊗ q^{∗}](f) = q^{∗}(Δ(p, s^{∗})_{Q}(fq))

uniquely determines the operator Δ_{Q}.

6.2. The horizontal de Rham complex. Consider a complex of C-dif-ferential operators ⋯ ⟶ P_{i−1} ⟶^{Δ_{i}} P_{i} ⟶^{Δ_{i+1}} P_{i+1} ⟶ · · ·. Multiplying it by a C-module Q and taking into account Proposition 6.2 on the preceding page, we obtain the complex

⋯ ⟶ P_{i−1} ⊗ Q ⟶^{(Δ_{i})_{Q}} P_{i} ⊗ Q ⟶^{(Δ_{i+1})_{Q}} P_{i+1} ⊗ Q ⟶ · · · .

Applying this construction to the horizontal de Rham complex, we get horizontal de Rham complex with coefficients in Q:

0 ⟶ Q ⟶^{d̄_{Q}} Λ̄^{1} ⊗_{F} Q ⟶^{d̄_{Q}} ⋯ ⟶^{d̄_{Q}} Λ̄^{n} ⊗_{F} Q ⟶ 0,

where Λ̄^{i} = Λ̄^{i}(E).

The cohomology of the horizontal de Rham complex with coefficients in Q is said to be horizontal cohomology and is denoted by H̄^{i}(Q).

Exercise 6.4. Proof that the differential d̄= d̄_{Q} can also be defined by

(d̄q)(X) = ∇_{X}(q), q ∈ Q,
d̄(ω ⊗ q) = d̄ω ⊗ q + (−1)^{p}ω ∧ d̄q, ω ∈ Λ̄^{p}.

One easily sees that a morphism f: Q_{1} → Q_{2} of C-modules gives rise to a cochain mapping of the de Rham complexes:

0 ⟶ Q_{1} ⟶^{d̄} Λ̄^{1} ⊗_{F} Q_{1} ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n} ⊗_{F} Q_{1} ⟶ 0
↓  ↓  ↓
0 ⟶ Q_{2} ⟶^{d̄} Λ̄^{1} ⊗_{F} Q_{2} ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n} ⊗_{F} Q_{2} ⟶ 0.

Let us discuss some examples of horizontal de Rham complexes.

Example 6.9. The horizontal de Rham complex with coefficients in the module J̄^{∞}(P)

0 ⟶ J̄^{∞}(P) ⟶^{d̄} Λ̄^{1} ⊗ J̄^{∞}(P) ⟶^{d̄} Λ̄^{2} ⊗ J̄^{∞}(P) ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n} ⊗ J̄^{∞}(P) ⟶ 0

is the project limit of the horizontal Spencer complexes

0 ⟶ J̄^{k}(P) ⟶^{S̄} Λ̄^{1} ⊗ J̄^{k−1}(P) ⟶^{S̄} Λ̄^{2} ⊗ J̄^{k−2}(P) ⟶^{S̄} ⋯,  (6.4)

where S̄(ω ⊗ j̄_{l}(p)) = d̄ω ⊗ j̄_{l−1}(p). As usual Spencer complexes, they are exact in positive degrees and

H^{0}(Λ̄^{•} ⊗ J̄^{k−•}(P)) = P.

Recall that one proves this fact by considering the commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶  S̄^{k} ⊗ P  ⟶  J̄^{k}(P)  ⟶ J̄^{k−1}(P) ⟶ 0
↓δ̄  ↓S̄  ↓S̄
0 ⟶ Λ̄^{1} ⊗ S̄^{k−1} ⊗ P ⟶ Λ̄^{1} ⊗ J̄^{k−1}(P) ⟶ Λ̄^{1} ⊗ J̄^{k−2}(P) ⟶ 0
↓δ̄  ↓S̄  ↓S̄
0 ⟶ Λ̄^{2} ⊗ S̄^{k−2} ⊗ P ⟶ Λ̄^{2} ⊗ J̄^{k−2}(P) ⟶ Λ̄^{2} ⊗ J̄^{k−3}(P) ⟶ 0
↓δ̄  ↓S̄  ↓S̄
⋮  ⋮  ⋮

(see page 20).

Exercise 6.5. Multiply this diagram by a C-module Q (possibly of infinite rank) and prove that the complex

0 ⟶ J̄^{∞}(P) ˆ⊗ Q ⟶^{d̄} Λ̄^{1} ⊗ J̄^{∞}(P) ˆ⊗ Q ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n} ⊗ J̄^{∞}(P) ˆ⊗ Q ⟶ 0

is exact in positive degrees and

H^{0}(Λ̄^{•} ⊗ J̄^{∞}(P) ˆ⊗ Q) = P ⊗ Q.

Here

J̄^{∞}(P) ˆ⊗ Q = proj lim J̄^{k}(P) ⊗ Q.

Example 6.10. The dualization of the previous example is as follows. The coefficient module is CDiff(P, F). The corresponding horizontal de Rham complex multiplied by a C-module Q has the form

0 ⟶ CDiff(P, F) ⊗ Q ⟶^{d̄} CDiff(P, Λ̄^{1}) ⊗ Q ⟶^{d̄} ⋯

⋯ ⟶^{d̄} CDiff(P, Λ̄^{n}) ⊗ Q ⟶ 0.

As in the previous example, it is easily shown that

H^{i}(CDiff(P, Λ̄^{•}) ⊗ Q) = 0 for i < n,
H^{n}(CDiff(P, Λ̄^{•}) ⊗ Q) = P̂ ⊗ Q,

where P̂ = Hom_{F}(P, Λ̄^{n}).

One can use this fact to define the notion of adjoint C-differential operator similarly to Definition 2.1 on page 27. The analog of Proposition 2.1 on page 27 remains valid for C-differential operators.

Example 6.11. Take the C-module

Q = ⨁_{p} D^{v}(C^{p}Λ) = ⨁_{p} Hom_{F}(C^{1}Λ, C^{p}Λ).

The horizontal de Rham complex with coefficients in Q can be written as

0 ⟶ D^{v} ⟶ D^{v}(Λ^{1}) ⟶ D^{v}(Λ^{2}) ⟶ ⋯

Proposition 6.3. The differential d_{D^{v}(C^{p}Λ)} of this complex is equal to −∂_{C} (see page 87), so that the complex coincides up to sign with the complex (5.39) on page 87.

Proof. Take a vertical vector field Y ∈ D^{v} and an arbitrary vector field Z. By (5.22) on page 81 we obtain (cf. the proof of Theorem 5.12 on page 87) i_{Z}∂_{C}Y = [Z^{v} − Z, Y]^{v}. Hence, ∂_{C}(D^{v}) ⊂ D^{v} ⊗ Λ̄^{1} and ∂_{C}|_{D^{v}} = −d_{D^{v}}. This together with formula (5.37) on page 85 and Remark 5.5 on page 87 yields ∂_{C}(D^{v}(C^{p}Λ) ⊗ Λ̄^{q}) ⊂ D^{v}(C^{p}Λ) ⊗ Λ̄^{q+1} and ∂_{C}|_{D^{v}(C^{p}Λ)⊗Λ̄^{q}} = −d_{D^{v}(C^{p}Λ)}.

6.3. Horizontal compatibility complex. Consider a C-differential operator Δ: P_{0} → P_{1}. It is clear that by repeating word by word the construction of Subsection 1.4 on page 13 one obtains the horizontal compatibility complex

P_{0} ⟶^{Δ} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3} ⟶^{Δ_{3}} ⋯,  (6.5)

which is formally exact (see the end of Subsection 1.7 on page 25).

Consider the C-module R_{Δ} = ker ψ_{∞}^{Δ} (cf. Example 6.6 on page 102). Then by Theorem 1.20 on page 21 the cohomology of complex (6.5) is isomorphic to the horizontal cohomology with coefficients in R_{Δ}:

Theorem 6.4.

H̄^{i}(R_{Δ}) = H^{i}(P_{•}).

Recall that this theorem follows from the spectral sequence arguments applied to the commutative diagram

⋮  ⋮  ⋮
↑  ↑  ↑
0 ⟶ Λ̄^{2} ⊗ J̄^{∞}(P_{0}) ⟶ Λ̄^{2} ⊗ J̄^{∞}(P_{1}) ⟶ Λ̄^{2} ⊗ J̄^{∞}(P_{2}) ⟶ ⋯
↑d̄  ↑d̄  ↑d̄
0 ⟶ Λ̄^{1} ⊗ J̄^{∞}(P_{0}) ⟶ Λ̄^{1} ⊗ J̄^{∞}(P_{1}) ⟶ Λ̄^{1} ⊗ J̄^{∞}(P_{2}) ⟶ ⋯
↑d̄  ↑d̄  ↑d̄
0 ⟶ J̄^{∞}(P_{0}) ⟶ J̄^{∞}(P_{1}) ⟶ J̄^{∞}(P_{2}) ⟶ ⋯
↑  ↑  ↑
0  0  0

Let us multiply this diagram by a C-module Q. This yields

H̄^{i}(R_{Δ} ˆ⊗ Q) = H^{i}(P_{•} ⊗ Q),  (6.6)

where R_{Δ} ˆ⊗ Q = proj lim R_{Δ}^{l} ⊗ Q, with R_{Δ}^{l} = ker ψ_{k+l}^{Δ}, ord Δ ≤ k.

We can dualize our discussion. Namely, consider the commutative diagram

⋮  ⋮  ⋮
↓  ↓  ↓
0 ⟵ CDiff(P_{0}, Λ̄^{n−2}) ⟵ CDiff(P_{1}, Λ̄^{n−2}) ⟵ CDiff(P_{2}, Λ̄^{n−2}) ⟵ ⋯
↓d̄  ↓d̄  ↓d̄
0 ⟵ CDiff(P_{0}, Λ̄^{n−1}) ⟵ CDiff(P_{1}, Λ̄^{n−1}) ⟵ CDiff(P_{2}, Λ̄^{n−1}) ⟵ ⋯
↓d̄  ↓d̄  ↓d̄
0 ⟵ CDiff(P_{0}, Λ̄^{n}) ⟵ CDiff(P_{1}, Λ̄^{n}) ⟵ CDiff(P_{2}, Λ̄^{n}) ⟵ ⋯
↓  ↓  ↓
0  0  0

As above, we readily obtain

H̄^{i}(R_{Δ}^{∗}) = H_{n−i}(P̂_{•})

and, more generally,

H̄^{i}(R_{Δ}^{∗} ⊗ Q) = H_{n−i}(P̂_{•} ⊗ Q),  (6.7)

where R_{Δ}^{∗} = Hom(R_{Δ}, F). The homology in the right-hand side of these formulae is the homology of the complex

P̂_{0} ⟵^{Δ^{∗}} P̂_{1} ⟵^{Δ_{1}^{∗}} P̂_{2} ⟵^{Δ_{2}^{∗}} P̂_{3} ⟵^{Δ_{3}^{∗}} ⋯,

dual to the complex (6.5).

6.4. Applications to computing the C-cohomology groups. Let E be an equation,

P_{0} = κ ⟶^{ℓ_{E}} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3} ⟶^{Δ_{3}} P_{4} ⟶^{Δ_{4}} ⋯

the compatibility complex for the operator of universal linearization, κ = F(E, π). Take a C-module Q.

Theorem 6.5. H̄^{i}(D^{v}(Q)) = H^{i}(P_{•} ⊗ Q).

Proof. The statement follows immediately from (6.6) on the page before and Proposition 3.30 on page 67.

Let Q = C^{p}Λ. The previous theorem gives a method for computing of the cohomology groups H̄^{i}(D^{v}(C^{p}Λ)), which are the C-cohomology groups (see Example 6.11 on page 106):

Corollary 6.6. H̄^{i}(D^{v}(C^{p}Λ)) = H^{i}(P_{•} ⊗ C^{p}Λ).

Let us describe the isomorphisms given by this corollary in an explicit form.

Consider an element ∑_{i ∈ I} ω_{i}^{q} ⊗ j̄_{∞}(s_{i}) ∈ Λ̄^{q} ⊗ D^{v}(C^{p}Λ), where ω_{i}^{q} ∈ Λ̄^{q} ⊗ C^{p}Λ, s_{i} ∈ κ, which is a horizontal cocycle. This means that

∑_{i ∈ I} ω_{i}^{q} ⊗ j̄_{∞}(ℓ_{E}(s_{i})) = 0 and ∑_{i ∈ I} d̄ω_{i}^{q} ⊗ j̄_{∞}(s_{i}) = 0.

From the second equality it easily follows that there exists an element ∑_{i ∈ I_{1}} ω_{i}^{q−1} ⊗ j̄_{∞}(s′_{i}) ∈ Λ̄^{q−1} ⊗ C^{p}Λ ⊗ J̄^{∞}(κ), such that ∑_{i ∈ I_{1}} d̄ω_{i}^{q−1} ⊗ j̄_{∞}(s′_{i}) = ∑_{i ∈ I} ω_{i}^{q} ⊗ j̄_{∞}(s_{i}). Denote s_{i}^{1} = ℓ_{E}(s′_{i}). The element ∑_{i ∈ I_{1}} ω_{i}^{q−1} ⊗ j̄_{∞}(s_{i}^{1}) ∈ Λ̄^{q−1} ⊗ C^{p}Λ ⊗ J̄^{∞}(P_{1}) satisfies

∑_{i ∈ I_{1}} ω_{i}^{q−1} ⊗ j̄_{∞}(Δ_{1}(s_{i}^{1})) = 0 and ∑_{i ∈ I_{1}} d̄ω_{i}^{q−1} ⊗ j̄_{∞}(s_{i}^{1}) = 0.

Continuing this process, we obtain elements ∑_{i ∈ I_{l}} ω_{i}^{q−l} ⊗ j̄_{∞}(s_{i}^{l}) ∈ Λ̄^{q−l} ⊗ C^{p}Λ ⊗ J̄^{∞}(P_{l}) such that

∑_{i ∈ I_{l}} ω_{i}^{q−l} ⊗ j̄_{∞}(Δ_{l}(s_{i}^{l})) = 0 and ∑_{i ∈ I_{l}} d̄ω_{i}^{q−l} ⊗ j̄_{∞}(s_{i}^{l}) = 0.

For l = q these formulae mean that the element ∑_{i ∈ I_{q}} ω_{i}^{0}⊗j̄_{∞}(s_{i}^{q}) represents an element of the module P_{q} ⊗ C^{p}Λ that lies in the kernel of the operator Δ_{q+1}. This is the element that gives rise to the cohomology class in the group H^{q}(P_{•} ⊗ C^{p}Λ) corresponding to the chosen element of Λ̄^{q} ⊗ D^{v}(C^{p}Λ).

It follows from our results that if there is an integer k such that P_{k} = P_{k+1} = P_{k+2} = ⋯ = 0, i.e., the compatibility complex has the form

P_{0} = κ ⟶^{ℓ_{E}} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3} ⟶^{Δ_{3}} ⋯ ⟶^{Δ_{k−2}} P_{k−1} ⟶ 0,

then

H^{i}(D^{v}(C^{p}Λ)) = 0 for i ≥ k.

This result is known as the k-line theorem for the C-cohomology.

What are the values of the integer k for differential equations encountered in mathematical physics? The existence of a compatibility operator Δ_{1} is usually due to the existence of dependencies between the equations under consideration: Δ_{1}(F) = 0, E = {F = 0}. The majority of systems that occur in practice consist of independent equations and for them k = 2. Such systems of differential equations are said to be ℓ-normal. In the case of ℓ-normal equations the two-line theorem for the C-cohomology holds:

Theorem 6.7 (the two-line theorem). Let a differential equation E be ℓnormal. Then:

(1) H^{i}(D^{v}(C^{p}Λ)) = 0 for i ≥ 2,

(2) H^{0}(D^{v}(C^{p}Λ)) = ker(ℓ_{E})_{C^{p}Λ},

(3) H^{1}(D^{v}(C^{p}Λ)) = coker(ℓ_{E})_{C^{p}Λ}.

Further, we meet with the case k > 2 in gauge theories, when the dependencies Δ_{1}(F) = 0 are given by the second Noether theorem (see page 126). For usual irreducible gauge theories, like electromagnetism, Yang - Mills models, and Einstein’s gravity, the Noether identities are independent, so that the operator Δ_{2} is trivial and, thus, k = 3. Finally, for an L-th stage reducible gauge theory, one has k = L + 3.

Remark 6.2. For the “empty” equation J^{∞}(π) Corollary 6.6 on the facing page yields Theorem 5.13 on page 88 (the one-line theorem).

6.5. Example: Evolution equations. Consider an evolution equation E = {F = u_{t} − f(x, t, u_{i}) = 0}, with independent variables x, t and dependent variable u; u_{i} denotes the set of variables corresponding to derivatives of u with respect to x.

Natural coordinates for E^{∞} are (x, t, u_{i}). The total derivatives operators D_{x} and D_{t} on E^{∞} have the form

D_{x} = \frac{∂}{∂x} + ∑_{i} u_{i+1} \frac{∂}{∂u_{i}} , D_{t} = \frac{∂}{∂t} + ∑_{i} D_{x}^{i}(f)\frac{∂}{∂u_{i}} .

The operator of universal linearization is given by

ℓ_{E} = D_{t} − ℓ_{f} = D_{t} − ∑_{i} \frac{∂f}{∂u_{i}} D_{x}^{i}.

Clearly, for an evolution equation the two-line theorem holds, hence the C-cohomology H̄^{q}(D^{v}(C^{p}Λ)) is trivial for q ≥ 2. Now, assume that the order of the equation E is greater than or equal to 2, i.e., ord ℓ_{f} ≥ 2. Then one has more:

Theorem 6.8. For any evolution equation of order ≥ 2, one has

H̄^{0}(D^{v}(C^{p}Λ)) = 0 for p ≥ 2,

Proof. It follows from Theorem 6.7 on the preceding page that H̄^{0}(D^{v}(C^{p}Λ)) = ker(ℓ_{E})_{C^{p}Λ}. Hence to prove the theorem it suffices to check that the equation

(D_{t} − ℓ_{f})(ω) = 0,  (6.8)

with ω ∈ C^{p}Λ, has no nontrivial solutions for p ≥ 2.

To this end consider the symbol of (6.8). Denote smbl(D_{x}) = θ. The

symbol of ℓ_{f} has the form smbl(ℓ_{f}) = gθ^{k}, k = ord ℓ_{f} ≥ 2, where g = \frac{∂f}{∂u_{k}}.

An element ω ∈ C^{p}Λ can be identified with a multilinear C-differential operator, so the symbol of ω is a homogeneous polynomial in p variables smbl(ω) = δ(θ_{1}, …, θ_{p}). Equation (6.8) yields

[g(θ_{1}^{k} + ⋯ + θ_{p}^{k}) − g(θ_{1} + ⋯ + θ_{p})^{k}] · δ(θ_{1}, …, θ_{p}) = 0.

The conditions k ≥ 2 and p ≥ 2 obviously imply that δ(θ_{1}, …, θ_{p}) = 0. This completes the proof.

Remark 6.3. This proof can be generalized for determined systems of evolution equations with arbitrary number of independent variables (see [16]).

7. Vinogradov’s C-spectral sequence

7.1. Definition of the Vinogradov C-spectral sequence. Suppose E ⊂ J^{k}(π) is a formally integrable differential equation. Consider the ideal CΛ^{∗} = CΛ^{∗}(E) of the exterior algebra Λ^{∗}(E) of differential forms on E^{∞} generated by the Cartan submodule C^{1}Λ(E) (see page 60): CΛ^{∗} = C^{1}Λ(E) ∧ Λ^{∗}(E). Clearly, this ideal and all its powers (CΛ^{∗})^{∧s} = C^{s}Λ ∧ Λ^{∗}, where C^{s}Λ = \underbrace{C^{1}Λ ∧ ⋯ ∧ C^{1}Λ}_{s times}, is stable with respect to the operator d, i.e.,

d((CΛ^{∗})^{∧s}) ⊂ (CΛ^{∗})^{∧s}.

Thus, in the de Rham complex on E^{∞} we have the filtration

Λ^{∗} ⊃ CΛ^{∗} ⊃ (CΛ^{∗})^{∧2} ⊃ ⋯ ⊃ (CΛ^{∗})^{∧s} ⊃ · · · .

The spectral sequence (E_{r}^{p,q}, d_{r}^{p,q}) determined by this filtration is said to be the Vinogradov C-spectral sequence of equation E. As usual p is the filtration degree and p + q is the total degree.

It follows from the direct sum decomposition (3.40) on page 60 that E_{0}^{p,q} can be identified with C^{p}Λ ⊗ Λ̄^{q}.

Exercise 7.1. Prove that under this identification the operator d_{0}^{p,q} coincides with the horizontal de Rham differential d̄_{C^{p}Λ} with coefficients in C^{p}Λ (cf. Example 6.3 on page 101).

Thus, the Vinogradov C-spectral sequence is one of two spectral sequences associated with the variational bicomplex (C^{p}Λ ⊗ Λ̄^{q}), d̄, d_{C}) constructed in Subsection 3.8 on page 60.

Remark 7.1. The second spectral sequences associated with the variational bicomplex can be naturally identified with the Leray–Serre spectral sequence of the de Rham cohomology of the bundle E^{∞} → M.

Remark 7.2. The definition of the Vinogradov C-spectral sequence given above remains valid for any object the category Inf (see page 68), whereas the variational bicomplex exists only for an infinite prolonged equation.

Exercise 7.2. Prove that any morphism F: N_{1} → N_{2} in Inf gives rise to the homomorphism of the Vinogradov C-spectral sequence for N_{2} into the Vinogradov C-spectral sequence for N_{1}.

7.2. The term E_{1} for J^{∞}(π). Let us consider the term E_{1} of the Vinogradov C-spectral sequence for the “empty” equation E^{∞} = J^{∞}(π).

By definition the first term E_{1} of a spectral sequence is the cohomology of its zero term E_{0}. Thus, to describe the terms E_{1}^{p,q}(π) we must compute the cohomologies of complexes

0 ⟶ C^{p}Λ(π) ⟶^{d̄} C^{p}Λ(π) ⊗ Λ̄^{1}(π) ⟶^{d̄} ⋯ ⟶^{d̄} C^{p}Λ(π) ⊗ Λ̄^{n}(π) ⟶ 0.

Using Proposition 3.30 on page 67, this complex can be rewritten in the form

0 ⟶ CDiff_{(p)}^{alt}(κ(π), F(π)) ⟶^{w} CDiff_{(p)}^{alt}(κ(π), Λ̄^{1}(π)) ⟶^{w} ⋯

⟶^{w} CDiff_{(p)}^{alt}(κ(π), Λ̄^{n}(π)) ⟶ 0,

where w(Δ) = (−1)^{p}d̄∘ Δ.

Now from Theorem 2.8 on page 32 we obtain the following description of the term E_{1} for J^{∞}(π):

Theorem 7.1. Let π be a smooth vector bundle over a manifold M, dim M = n. Then:

(1) E_{1}^{0,q}(π) = H̄^{q}(π) for all q ≥ 0;

(2) E_{1}^{p,q}(π) = 0 for p > 0, q ≠ n;

(3) E_{1}^{p,n}(π) = L_{p}^{alt}(κ(π)), p > 0,

where L_{p}^{alt}(κ(π)) was defined in Theorem 2.8 on page 32.

Since the the Vinogradov C-spectral sequence converges to the de Rham cohomology of the manifold J^{∞}(π), this theorem has the following

Corollary 7.2. For any smooth vector bundle π over an n-dimensional smooth manifold M one has:

(1) E_{r}^{p,q}(π) = 0, 1 ≤ r ≤ ∞, if p > 0, q ≠ n or p = 0, q > n;

(2) E_{1}^{0,q}(π) = E_{∞}^{0,q}(π) = H^{q}(J^{∞}(π)) = H^{q}(J^{0}(π)), q < n;

(3) E_{2}^{p,n}(π) = E_{∞}^{p,n}(π) = H^{p+n}(J^{∞}(π)) = H^{p+n}(J^{0}(π)), p ≥ 0.

Exercise 7.3. Prove that H^{q}(J^{∞}(π)) = H^{q}(J^{0}(π)).

We now turn our attention to the differentials d_{1}^{p,n}. They are induced by the Cartan differential d_{C}. For p = 0, we have d_{C}(ω) = ℓ_{ω}, ω ∈ Λ̄^{n}. (Note that the expression ℓ_{ω} is correct, because ω is a horizontal form, i.e., a nonlinear operator from Γ(π) to Λ^{n}(M).) Therefore the operator

E_{1}^{0,n}(π) = H̄^{n}(π) ⟶^{d_{1}^{0,n}} E_{1}^{1,n}(π) = κ̂(π)

is given by the formula d_{1}^{0,n}([ω]) = μ(ℓ_{ω}) = ℓ_{ω}^{∗}(1), where ω ∈ Λ̄^{n}(π), [ω] is the horizontal cohomology class of ω.

Exercise 7.4. Write down the coordinate expression for the operator d_{1}^{0,n} and show that it coincides with the standard Euler operator, i.e., with the operator that takes a Lagrangian to the corresponding Euler–Lagrange equation.

Let us compute the operators d_{1}^{p,n}, p > 0.

Consider an element ∇ ∈ L_{p}^{alt}(κ(π)) and define the operator □ ∈ CDiff_{(p+1)}(κ(π), Λ̄^{n}(π)) via

□(χ_{1}, …, χ_{p+1}) = ∑_{i=1}^{p+1}(−1)^{i+1}�_{χ_{i}}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p+1}))

+ ∑_{1≤i<j≤p+1}(−1)^{i+j}∇({χ_{i}, χ_{j}}, χ_{1}, …, χ̂_{i}, …, χ̂_{j}, …, χ_{p+1}). (7.1)

Exercise 7.5. Prove that d_{1}^{p,n}(∇) = μ_{(p+1)}(□) (see page 33 for the definition of μ_{(p+1)}).

Remark 7.3. Needless to say that this fact follows from the standard formula for exterior differential. It needs however to be proved that one may use this formula even though ∇ as an element of CDiff_{(p)}(κ, Λ̄^{n}) is not skew-symmetric.

From (7.1) we get

□(χ_{1}, …, χ_{p+1}) = ∑_{i=1}^{p}(−1)^{i+1}�_{χ_{i}}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p}))(χ_{p+1})

+ ∑_{i=1}^{p}(−1)^{i+1}∇(χ_{1}, …, χ̂_{i}, …, χ_{p}, �_{χ_{i}}(χ_{p+1}))

+ (−1)^{p}�_{χ_{p+1}}(∇(χ_{1}, …, χ_{p}))

+ ∑_{1≤i<j≤p}(−1)^{i+j}∇({χ_{i}, χ_{j}}, χ_{1}, …, χ̂_{i}, …, χ̂_{j}, …, χ_{p+1})

+ ∑_{i=1}^{p}(−1)^{i+p+1}∇({χ_{i}, χ_{p+1}}, χ_{1}, …, χ̂_{i}, …, χ_{p})

= ∑_{i=1}^{p}(−1)^{i+1}�_{χ_{i}}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p}))(χ_{p+1})

+ ∑_{1≤i<j≤p}(−1)^{i+j}∇({χ_{i}, χ_{j}}, χ_{1}, …, χ̂_{i}, …, χ̂_{j}, …, χ_{p+1})

+ ∑_{i=1}^{p}(−1)^{i+1}∇(χ_{1}, …, χ̂_{i}, …, χ_{p}, ℓ_{χ_{i}}(χ_{p+1})) + (−1)^{p}ℓ_{∇(χ_{1},...,χ_{p})}(χ_{p+1}).

Therefore

d_{1}^{p,n}(∇)(χ_{1}, …, χ_{p}) = μ_{(p+1)}(□)(χ_{1}, …, χ_{p})

= ∑_{i=1}^{p}(−1)^{i+1}�_{χ_{i}}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p}))

+ ∑_{i<j}(−1)^{i+j}∇({χ_{i}, χ_{j}}, χ_{1}, …, χ̂_{i}, …, χ̂_{j}, …, χ_{p})

+ ∑_{i=1}^{p}(−1)^{i+1}ℓ_{χ_{i}}^{∗}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p})) + (−1)^{p}ℓ_{∇(χ_{1},...,χ_{p})}^{∗}(1). (7.2)

Exercise 7.6. Prove that

ℓ_{ψ(φ)}^{∗}(1) = ℓ_{ψ}^{∗}(φ) + ℓ_{φ}^{∗}(ψ), φ ∈ κ(π), ψ ∈ κ̂(π). Using this formula, let us rewrite the last term of (7.2) in the following

way:

(−1)^{p}ℓ_{(∇(χ_{1},...,χ_{p}))}^{∗}(1) = \frac{1}{p} ∑_{i=1}^{p}(−1)^{i}(ℓ_{(∇(χ_{1},...,χ̂_{i},...,χ_{p},χ_{i}))}^{∗}(1)

= \frac{1}{p} ∑_{i=1}^{p}(−1)^{i}(ℓ_{∇(χ_{1},...,χ̂_{i},...,χ_{p})}^{∗}(χ_{i}) + ℓ_{χ_{i}}^{∗}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p}))).

Finally we obtain

(d_{1}^{p,n}(∇)) (χ_{1}, …, χ_{p}) = ∑_{i=1}^{p}(−1)^{i+1}�_{χ_{i}}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p}))

+ ∑_{i<j}(−1)^{i+j}∇({χ_{i}, χ_{j}}, χ_{1}, …, χ̂_{i}, …, χ̂_{j}, …, χ_{p})

+ \frac{1}{p} ∑_{i=1}^{p}(−1)^{i+1}((p − 1)ℓ_{χ_{i}}^{∗}(∇(χ_{1}, …, χ̂_{i}, …, χ_{p})) − ℓ_{∇(χ_{1},...,χ̂_{i},...,χ_{p})}^{∗}(χ_{i})).

In particular, for p = 1 we have d_{1}^{1,n}(ψ)(φ) = �_{φ}(ψ) − ℓ_{ψ}^{∗}(φ) = ℓ_{ψ}(φ) − ℓ_{ψ}^{∗}(φ), ψ ∈ κ̂(π), φ ∈ κ(π), that is

d_{1}^{1,n}(ψ) = ℓ_{ψ} − ℓ_{ψ}^{∗}.

Consider the following complex, which is said to be the (global) variational complex,

0 ⟶ F(π) ⟶^{d̄} Λ̄^{1}(π) ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n}(π) ⟶^{E} E_{1}^{1,n}(π) ⟶^{d_{1}^{1,n}} E_{1}^{2,n}(π) ⟶^{d_{1}^{2,n}} ⋯,

where operator E is equal to the composition of the natural projection

Λ̄^{n}(π) → H̄^{n}(π) and the operator H̄^{n}(π) ⟶^{d_{1}^{0,n}} E_{1}^{1,n}(π) ^{15}.

In view of Corollary 7.2 on page 112, the cohomology of this complex coincides with H^{∗}(J^{0}(π)).

The operator E is the Euler operator (see Exercise 7.4 on page 112). It takes each Lagrangian density ω ∈ Λ_{0}^{n}(π) to the left-hand part of the corresponding Euler–Lagrange equation E(ω) = 0. Thus the action functional

s ↦ ∫_{M} j_{∞}(s)^{∗}(ω), s ∈ Γ(π),

is stationary on the section s if and only if j_{∞}(s)^{∗} (E(ω)) = 0.

If the cohomology of the space J^{0}(π) is trivial, then the variational complex is exact. This immediately implies a number of consequences. The three most important are:

(1) ker E = im d̄(“a Lagrangian with zero variational derivative is a total divergence”);

(2) d̄ω = 0 if and only if ω is of the form ω = d̄η, ω ∈ Λ̄^{n−1}(π) (“all zero total divergence are total curls”);

(3) ℓ_{ψ} = ℓ_{ψ}^{∗} if and only if ψ is of the form ψ = E(ω), ψ ∈ κ̂(π) (this is the solution of the inverse problem to the calculus of variations).

Now suppose that we are given ψ ∈ κ̂(π) such that ℓ_{ψ} = ℓ_{ψ}^{∗}. How one can find a Lagrangian ω such that ψ = E(ω)? To this end take a oneparameter family of fiberwise transformations G_{t}: E → E, 0 ≤ t ≤ 1, of the space of the bundle π: E → M, with G_{0} = 0 and G_{1} = id_{E}. Consider the corresponding family of evolutionary vector field �_{φ_{t}}, i.e.,

\frac{d}{dt}G_{t}^{(∞)∗} = �_{φ_{t}} ∘ G_{t}^{(∞)∗}

for t > 0. Let us compute the correspondent Lie derivative �_{φ_{t}}(ψ) (which is different from the usual “component-wise” derivative). Take Ω ∈ Λ̄^{n} ⊗ C^{1}Λ, dΩ = 0, that represents ψ. Then �_{φ_{t}}(Ω) = d_{C}(Ω(�_{φ_{t}})). Therefore �_{φ_{t}}(ψ) = E(ψ(φ_{t})). Hence

\frac{d}{dt}G_{t}^{(∞)∗}(ψ) = E(G_{t}^{(∞)∗}(ψ(φ_{t}))),

and integrating this with respect to t, we obtain the following homotopy (or inverse) formula

ψ = E (∫_{0}^{1} G_{t}^{(∞)∗}(ψ(φ_{t})) dt) .

15Below we use the notation E for the operator d_{1}^{0,n}: H̄^{n}(π) → E_{1}^{1,n}(π) as well.

Take, for instance, G_{t}(e_{x}) = te_{x}, e_{x} ∈ E_{x} = π^{−1}(x). Then φ_{t}^{i} = \frac{u^{i}}{t} and we have

ψ = E (∫_{0}^{1} ∑_{i} u^{i}ψ^{i}(x, tu_{σ}^{j}) dt) .

Exercise 7.7. Let Δ ∈ CDiff(P, Λ̄^{n}(π)). Using the Green formula and Exercise 7.6 on page 114, prove that for any p ∈ P one has

E(Δ(p)) = ℓ_{p}^{∗}(Δ^{∗}(1)) + ℓ_{Δ^{∗}(1)}^{∗}(p).

Deduce from this formula that for any φ ∈ κ(π) and ω ∈ Λ̄^{n}(π) the following equality holds

E(�_{φ}(ω)) = �_{φ}(E(ω)) + ℓ_{φ}^{∗}(E(ω)).

Exercise 7.8. Let J = (J_{0}, J_{1}, …, J_{n}) be a conserved current for an evolution equation E = {u_{t} = f(t, x, u, u_{x}, u_{xx}, …)}. Using the previous exercise, prove that the vector-function ψ = E(J_{0}), where J_{0} is the t-component of the conserved current that is regarded as a function of (t, x, u, u_{x}, u_{xx}, …), satisfies the equation

D_{t}(ψ) + ℓ_{f}^{∗}(ψ) = 0

(cf. Theorem 7.11 on page 122).

7.3. The term E_{1} for an equation. Let E be an equation,

P_{0} = κ ⟶^{ℓ_{E}} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3} ⟶^{Δ_{3}} P_{4} ⟶^{Δ_{4}} ⋯

be the compatibility complex for the universal linearization operator, and

P̂_{0} = κ̂ ⟵^{ℓ_{E}^{∗}} P̂_{1} ⟵^{Δ_{1}^{∗}} P̂_{2} ⟵^{Δ_{2}^{∗}} P̂_{3} ⟵^{Δ_{3}^{∗}} P̂_{4} ⟵^{Δ_{4}^{∗}} ⋯

be the dual complex. Take a C-module Q.

Theorem 7.3. For any equation E and a C-module Q one has

H̄^{n−i}(C^{1}Λ ⊗ Q) = H_{i}(P̂_{•} ⊗ Q).

Proof. The statement follows immediately from (6.7) on page 107 and Proposition 3.30 on page 67.

Let Q = C^{p}Λ. The theorem gives a method for computing the Vinogradov C-spectral sequence. Namely, since the term E_{1}^{p,q} = H̄^{q}(C^{p}Λ) of the Vinogradov C-spectral sequence is a direct summand in the cohomology group H̄^{q}(C^{1}Λ⊗C^{p−1}Λ), we have a description for the first term of the Vinogradov C-spectral sequence. Thus:

Corollary 7.4. The term E_{1}^{p,q} of the Vinogradov C-spectral sequence is the skew-symmetric part of the group H_{n−q}(P̂_{•} ⊗ C^{p−1}Λ).

It is useful to describe the isomorphisms given by this corollary in an explicit form.

Consider an operator ∇ ∈ CDiff(κ, Λ̄^{q} ⊗ C^{p−1}Λ) that represents an element of E_{1}^{p,q}. This means that

d̄∘ ∇ = ∇_{1} ∘ ℓ_{E}

for an operator ∇_{1} ∈ CDiff(P_{1}, Λ̄^{q+1} ⊗ C^{p−1}Λ). Applying the operator d̄ to both sides of this formula and using Exercise 1.3 on page 13, we get

d̄∘ ∇_{1} = ∇_{2} ∘ Δ_{1}

for some operator ∇_{2} ∈ CDiff(P_{2}, Λ̄^{q+2} ⊗ C^{p−1}Λ). Continuing this process, we obtain operators ∇_{i} ∈ CDiff(P_{i}, Λ̄^{q+i} ⊗ C^{p−1}Λ), i = 1, 2, …, n − q, such that

d̄∘ ∇_{i−1} = ∇_{i} ∘ Δ_{i−1}.

For i = n−q, this formula means that the operator ∇_{n−q} ∈ CDiff(P_{n−q}, Λ̄^{n}⊗ C^{p−1}Λ) represents an element of the module P̂_{n−q} ⊗ C^{p−1}Λ that lies in the kernel of the operator Δ_{n−q−1}^{∗}. This is the element that gives rise to the homology class in H_{n−q}(P̂_{•} ⊗ C^{p−1}Λ) corresponding to the chosen element of E_{1}^{p,q}.

If the compatibility complex has the length k,

P_{0} = κ ⟶^{ℓ_{E}} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3} ⟶^{Δ_{3}} ⋯ ⟶^{Δ_{k−2}} P_{k−1} ⟶ 0,

then E_{1}^{p,q} = 0 for p > 0 and q ≤ n − k. This is the k-line theorem for the Vinogradov C-spectral sequence.

In the case k = 2, i.e., for ℓ-normal equations, the two-line theorem holds:

Theorem 7.5 (the two-line theorem). Let E be an ℓ-normal differential equation. Then:

(1) E_{1}^{p,q} = 0 for p > 0 and q ≤ n − 2,

(2) E_{1}^{p,n−1} ⊂ ker(ℓ_{E}^{∗})_{C^{p−1}Λ} for p > 0,

(3) E_{1}^{p,n} ⊂ coker(ℓ_{E}^{∗})_{C^{p−1}Λ} for p > 0.

This theorem has the following elementary

Corollary 7.6. The terms E_{r}^{p,q}(E) of the Vinogradov C-spectral sequence satisfy the following:

(1) E_{r}^{p,q}(E) = 0 if p ≥ 1, q ≠ n − 1, n, 1 ≤ r ≤ ∞;

(2) E_{3}^{p,q}(E) = E_{∞}^{p,q}(E);

(3) E_{1}^{0,q}(E) = E_{∞}^{0,q}(E) = H^{q}(E^{∞}), q ≤ n − 2;

(4) E_{2}^{0,n−1}(E) = E_{∞}^{0,n−1}(E) = H^{n−1}(E^{∞});

(5) E_{2}^{1,n−1}(E) = E_{∞}^{1,n−1}(E).

Example 7.1. For an evolution equation E = {F = u_{t} − f(x, t, u, u_{x}, u_{xx} …) = 0} the two-line theorem implies that the Vinogradov C-spectral sequence is trivial for q ≠ 1, 2, p > 0, and exactly as in Example 6.5 on page 109 one proves that E_{1}^{p,1} = 0 for p ≥ 3.

7.4. Example: Abelian p-form theories. Let M be a (pseudo-)Rieman-nian manifold and π: E → M the p-th exterior power of the cotangent bundle over M, so that a section of π is a p-form on M. Evidently, on the jet space J^{∞}(π) there exists a unique horizontal form A ∈ Λ̄^{p}(J^{∞}(π)) such that j_{∞}^{∗}(ω)(A) = ω for all ω ∈ Λ^{p}(M). Consider the equation E = {F = 0}, with F = d̄∗d̄A, where ∗ is the Hodge star operator. Our aim is to calculate the terms of the Vinogradov C-spectral sequence E_{1}^{i,q}(E) for q ≤ n − 2. We shall assume that 1 ≤ p < n − 1 and that the manifold M is topologically trivial.

Obviously, we have P_{0} = κ = Λ̄^{p}, P_{1} = Λ̄^{n−p}, and ℓ_{E} = d̄∗d̄: Λ̄^{p} → Λ̄^{n−p}. Taking into account Example 1.2 on page 24, we see that the compatibility complex for ℓ_{E} has the form

\begin{matrix}Λ̄^{p} & ⟶^{ℓ_{E}} & Λ̄^{n−p} & ⟶^{d̄} & Λ̄^{n−p+1} & ⟶^{d̄} & ⋯ & ⟶^{d̄} & Λ̄^{n} & ⟶ & 0 \\ ∥ & & ∥ & & ∥ & & & & ∥ \\ P_{0} & & P_{1} & & P_{2} & & & & P_{k−1} \end{matrix} (7.3)

Thus k = p + 2 and the k-line theorem yields E_{1}^{i,q} = 0 for i > 0 and q < n − p − 1. Since the Vinogradov C-spectral sequence converges to the de Rham cohomology of E^{∞}, which is trivial, we also get E_{1}^{0,q} = 0 for 0 < q < n − p − 1, and dim E_{1}^{0,0} = 1, i.e., H̄^{1} = H̄^{2} = ⋯ = H̄^{n−p−2} = 0 and dim H̄^{0} = 1. Next, consider the terms E_{1}^{i,q} for n − p − 1 ≤ q < 2(n − p − 1) and i > 0. In view of Corollary 7.4 on page 116 one has

E_{1}^{i,q} ⊂ H̄^{q−(n−p−1)}(C^{i−1}Λ) = E_{1}^{i−1,q−(n−p−1)},

because the complex dual to the compatibility complex (7.3) has the form

Λ̄^{n−p} ⟵^{ℓ_{E}^{∗}} Λ̄^{p} ⟵^{d̄} Λ̄^{p−1} ⟵^{d̄} ⋯ ⟵^{d̄} F ⟵ 0.
∥  ∥  ∥  ∥
P̂_{0}  P̂_{1}  P̂_{2}  P̂_{p+1}

(Throughout, it is assumed that q ≤ n − 2.) Thus we obtain E_{1}^{i,q} = 0 for n − p − 1 < q < 2(n − p − 1), i > 0 and dim E_{1}^{1,n−p−1} = 1. Again, taking into account that the spectral sequence converges to the trivial cohomology, we get E_{1}^{0,q} = 0 for n − p − 1 < q < 2(n − p − 1) and dim E_{1}^{0,n−p−1} = 1. In addition, the map d_{1}^{0,n−p−1} : E_{1}^{0,n−p−1} → E_{1}^{1,n−p−1} is an isomorphism. Explicitly, one readily obtains that the one-dimensional space E_{1}^{0,n−p−1} is

[figure: n − p − 1 is even; n − p − 1 is odd; q; i; q; i; 3(n − p − 1); 2(n − p − 1); n − p − 1; 3(n − p − 1); 2(n − p − 1); n − p − 1]

Diagram 7.1

generated by the element ∗d̄A ∈ Λ̄^{n−p−1} and the map d_{1}^{0,n−p−1} takes this element to the operator ∗d̄: κ = Λ̄^{p} → Λ̄^{n−p−1}, which generates the space E_{1}^{1,n−p−1}.

Further, let us consider the terms E_{1}^{i,q} for 2(n − p − 1) ≤ q < 3(n − p − 1). Arguing as before, we see that all these terms vanish unless q = 2(n−p−1) and i = 0, 1, 2, with dim E_{1}^{1,2(n−p−1)} = 1 and dim E_{1}^{i,2(n−p−1)} ≤ 1, i = 0, 2. To compute the terms E_{1}^{i,2(n−p−1)} for i = 0 and i = 2, we have to consider two cases: n − p − 1 is even and n − p − 1 is odd (see Diagram 7.1).

In the first case, the map d_{1}^{1,2(n−p−1)} : E_{1}^{1,2(n−p−1)} → E_{1}^{2,2(n−p−1)} is trivial. Indeed, the operator (∗d̄A) ∧ ∗d̄: κ = Λ̄^{p} → Λ̄^{2(n−p−1)}, which generates the space E_{1}^{1,2(n−p−1)}, under the mapping d_{1}^{1,2(n−p−1)} is the antisymmetrization of the operator (ω_{1}, ω_{2}) ↦ (∗d̄ω_{1}) ∧ (∗d̄ω_{2}), ω_{i} ∈ κ = Λ̄^{p}. But this operator is symmetric, so that d_{1}^{1,2(n−p−1)} = 0. Consequently, E_{1}^{2,2(n−p−1)} = 0 and dim E_{1}^{0,2(n−p−1)} = 1. This settles the case when n − p − 1 is even.

In the case when n−p−1 is odd, the operator (ω_{1}, ω_{2}) ↦ (∗d̄ω_{1})∧(∗d̄ω_{2}) is skew-symmetric, hence the map d_{1}^{1,2(n−p−1)} is an isomorphism. Thus, dim E_{1}^{2,2(n−p−1)} = 1 and E_{1}^{0,2(n−p−1)} = 0.

Continuing this line of reasoning, we obtain the following result.

Theorem 7.7. For i = q = 0 one has dim E_{1}^{0,0} = 1. If either or both i and q are positive, there are two cases:

(1) if n − p − 1 is even then

dim E_{1}^{i,q} = \begin{cases}1 & for i = l(n − p − 1) and q = 0, 1, \\ 0 & otherwise; \end{cases}

(2) if n − p − 1 is odd then

dim E_{1}^{i,q} = \begin{cases}1 & for i = l(n − p − 1) and q = l − 1, l, \\ 0 & otherwise. \end{cases}

Here 1 ≤ l < \frac{n − 1}{n − p − 1}.

In other words, let Ā be the exterior algebra generated by two forms: ω_{1} = ∗d̄A ∈ Λ̄^{n−p−1} and ω_{2} = d̄_{1}(ω_{1}) = ∗d̄ ∈ Λ̄^{n−p−1} ⊗ C^{1}Λ; then we see that the space ⨁_{i,q≤n−2} E_{1}^{i,q} is isomorphic to the subspace of Ā containing no forms of degree q > n − 2.

7.5. Conservation laws and generating functions. We start by describing the differentials d_{1}^{0,n−1} and d_{1}^{1,n−1} for an ℓ-normal equation since they directly relate to the theory of conservation laws.

Suppose that an ℓ-normal equation E ⊂ J^{k}(π) is given by a section F ∈ F(π, ξ) = P.

Proposition 7.8. The operator

d_{1}^{0,n−1} : E_{1}^{0,n−1}(E) = H̄^{n−1}(E) → E_{1}^{1,n−1}(E) = ker (ℓ_{E})^{∗} ⊂ P̂

has the form

d_{1}^{0,n−1}(h) = □^{∗}(1),

where h = [ω] ∈ H̄^{n−1}(E), ω ∈ Λ̄^{n−1}(E) and □ ∈ CDiff(P, Λ̄^{n}(E)) is an operator satisfying d̄ω = □(F).

Proof. We have d̄∘ ℓ_{ω} = □ ∘ ℓ_{E}. Thus □ is an operator that represents the element d_{1}^{0,n−1}(h) ∈ E_{1}^{1,n−1}(E). Hence d_{1}^{0,n−1}(h) = □^{∗}(1).

Proposition 7.9. The term E_{1}^{2,n−1}(E) can be described as the quotient

{ ∇ ∈ CDiff(κ, P̂) | ℓ_{E}^{∗} ∘ ∇ = ∇^{∗} ∘ ℓ_{E} }/θ,

where θ = { □ ∘ ℓ_{E} | □ ∈ CDiff(P, P̂), □^{∗} = □ }.

Proof. Take a horizontal (n − 1)-cocycle with coefficients in C^{1}Λ ⊗ C^{1}Λ. Let an operator Δ ∈ CDiff(κ, P̂) corresponds to this cocycle by Theorem 7.3 on page 116. Then there exists an operator A ∈ CDiff(P, P̂) such that ℓ_{E}^{∗} ∘ Δ = A ∘ ℓ_{E}. By the Green formula we have

⟨ℓ_{E}^{∗}(Δ(χ_{1})), χ_{2}⟩ − ⟨Δ(χ_{1}), ℓ_{E}(χ_{2})⟩ = d̄(Δ_{1}(χ_{1}, χ_{2})),

where χ_{1}, χ_{2} ∈ κ, and Δ_{1} ∈ CDiff_{(2)}(κ, Λ^{n−1}). The cocycle under consideration belongs to E_{1}^{2,n−1}, if the operator Δ_{1} is skew-symmetric:

Δ_{1}(χ_{1}, χ_{2}) = −Δ_{1}(χ_{2}, χ_{1}) mod K,

where K ⊂ CDiff_{(2)}(κ, Λ̄^{n−1}) is the submodule consisting of the operators of the form γ(χ_{1}, χ_{2}) = γ_{1}(ℓ_{E}(χ_{1}), χ_{2}) + γ_{2}(ℓ_{E}(χ_{2}), χ_{1}) for some operators γ_{1}, γ_{2} ∈ CDiff(P, CDiff(κ, Λ̄^{n−1})). In this case

⟨ℓ_{E}^{∗}(Δ(χ_{1})), χ_{2}⟩ − ⟨Δ(χ_{1}), ℓ_{E}(χ_{2})⟩ = −⟨ℓ_{E}^{∗}(Δ(χ_{2})), χ_{1}⟩ + ⟨Δ(χ_{2}), ℓ_{E}(χ_{1})⟩ = −⟨ℓ_{E}^{∗}(Δ(χ_{2})), χ_{1}⟩ + ⟨χ_{2}, Δ^{∗}(ℓ_{E}(χ_{1}))⟩

= −⟨A(ℓ_{E}(χ_{2})), χ_{1}⟩ + ⟨χ_{2}, ℓ_{E}^{∗}(A^{∗}(χ_{1}))⟩

modulo d̄K. This implies Δ = A^{∗} + B ∘ ℓ_{E} for an operator B ∈ CDiff(P, P̂). One has ℓ_{E}^{∗} ∘ B ∘ ℓ_{E} = ℓ_{E}^{∗} ∘ Δ − ℓ_{E}^{∗} ∘ A^{∗} = ℓ_{E}^{∗} ∘ Δ − Δ^{∗} ∘ ℓ_{E}, hence B^{∗} = −B. Now we see that the operator ∇ = Δ − \frac{1}{2}B ∘ ℓ_{E} satisfies ℓ_{E}^{∗} ∘ ∇ = ∇^{∗} ∘ ℓ_{E}. The operator ∇ is defined modulo the operators of the form □ ∘ ℓ_{E}. We have ℓ_{E}^{∗} ∘ □ ∘ ℓ_{E} = ℓ_{E}^{∗} ∘ □^{∗} ∘ ℓ_{E}, so that □^{∗} = □.

Proposition 7.10. The operator d_{1}^{1,n−1} : E_{1}^{1,n−1}(E) = ker ℓ_{E}^{∗} → E_{1}^{2,n−1}(E) is given by

d_{1}^{1,n−1}(ψ) = (ℓ_{ψ} + Δ^{∗}) mod θ,

where Δ ∈ CDiff(P, κ̂) is an operator satisfying ℓ_{F}^{∗} (ψ) = Δ(F).

Proof. By Green’s formula on J^{∞}(π) we have

⟨ψ, ℓ_{F} (χ)⟩ − ⟨ℓ_{F}^{∗} (ψ), χ⟩ = d̄(□(χ)),

where χ ∈ κ(π), □ ∈ CDiff(κ(π), Λ̄^{n−1}(π)) = C^{1}Λ(π) ⊗ Λ̄^{n−1}(π). Let us compute d̄∘ d_{C}(□) ∈ C^{2}Λ(π) ⊗ Λ̄^{n}(π):

d̄(d_{C}(□)(χ_{1}, χ_{2})) = �_{χ_{1}}(d̄(□(χ_{2}))) − �_{χ_{2}}(d̄(□(χ_{1}))) − d̄(□({χ_{1}, χ_{2}})) = �_{χ_{1}}(⟨ψ, ℓ_{F} (χ_{2})⟩) − �_{χ_{2}}(⟨ψ, ℓ_{F} (χ_{1})⟩) − ⟨ψ, ℓ_{F} ({χ_{1}, χ_{2}})⟩

− �_{χ_{1}}(⟨ℓ_{F}^{∗} (ψ), χ_{2}⟩) + �_{χ_{2}}(⟨ℓ_{F}^{∗} (ψ), χ_{1}⟩) + ⟨ℓ_{F}^{∗} (ψ), {χ_{1}, χ_{2}}⟩

= ⟨ℓ_{ψ}(χ_{1}), ℓ_{F} (χ_{2})⟩ − ⟨ℓ_{ψ}(χ_{2}), ℓ_{F} (χ_{1})⟩ − ⟨ℓ_{Δ(F)}(χ_{1}), χ_{2}⟩ + ⟨ℓ_{Δ(F)}(χ_{2}), χ_{1}⟩.

Therefore, the restriction of d̄∘ d_{C}(□) to E^{∞} equals to

d̄∘ d_{C}(□)|_{E^{∞}} (χ_{1}, χ_{2}) = ⟨ℓ_{ψ}(χ_{1}), ℓ_{E}(χ_{2})⟩ − ⟨ℓ_{ψ}(χ_{2}), ℓ_{E}(χ_{1})⟩ − ⟨Δ(ℓ_{E}(χ_{1})), χ_{2}⟩ + ⟨Δ(ℓ_{E}(χ_{2})), χ_{1}⟩

= ⟨(ℓ_{ψ} + Δ^{∗})(χ_{1}), ℓ_{E}(χ_{2})⟩ − ⟨(ℓ_{ψ} + Δ^{∗})(χ_{2}), ℓ_{E}(χ_{1})⟩ + d̄γ(χ_{1}, χ_{2}),

where γ ∈ K. This completes the proof.

Now we apply these results to the problem of computing conservation laws of an ℓ-normal differential equation E.

First, note that for a formally integrable equation E the projections E^{(k+1)} → E^{(k)} are affine bundles, therefore E^{(k+1)} and E^{(k)} are of the same homotopy type. Hence, H^{∗}(E^{∞}) = H^{∗}(E).

Further, it follows from the two-line theorem that there exists the following exact sequence:

0 ⟶ H^{n−1}(E) ⟶ H̄^{n−1}(E) ⟶^{d_{1}^{0,n−1}} ker (ℓ_{E})^{∗} .

Recall that the group H̄^{n−1}(E) was interpreted as the group of conservation laws of the equation E (see the beginning of Section 6 on page 99). Conservation laws ω ∈ H^{n−1}(E) ⊂ H̄^{n−1}(E) are called topological (or rigid), since they are determined only by the topology of the equation E. In particular, the corresponding conserved quantities do not change under deformations of solutions of the equation E. Therefore topological conservation laws are not very interesting for us and we consider the quotient group cl(E) = H̄^{n−1}(E)/H^{n−1}(E), called the group of proper conservation laws of the equation E. The two-line theorem implies immediately the following.

Theorem 7.11. If E is an ℓ-normal equation, then

cl(E) ⊂ ker ℓ_{E}^{∗}.

If, moreover, H^{n}(E) ⊂ H̄^{n}(E) (in particular, H^{n}(E) = 0), we have

cl(E) = ker d_{1}^{1,n−1}.

Element ψ ∈ ker ℓ_{E}^{∗} that corresponds to a conservation law [ω] ∈ cl(E) is called its generating function.

Theorem 7.11 gives an effective method for computing conservation laws.

Remark 7.4. In view of Proposition 7.9 on page 120, elements of E_{1}^{2,n−1} can be interpreted as mappings from ker ℓ_{E}^{∗} to ker ℓ_{E}, i.e., from generating functions of conservation laws to symmetries of E.

Proposition 7.12. Let E = {u_{t} = f(t, x, u, u_{x}, u_{xx}, …)} be an evolution equation and J = (J_{0}, J_{1}, …, J_{n}) a conserved current for E. Then the generating function of J is equal to ψ = E(J_{0}), where J_{0} is the t-component of the conserved current that is regarded as a function of (t, x, u, u_{x}, u_{xx}, …).

Proof. The restriction of the total derivative D_{t} to the equation E^{∞} has the

form: D_{t} = \frac{∂}{∂t} + �_{f}. Hence \frac{∂J_{0}}{∂t} + �_{f}(J_{0}) + ∑_{i=1}^{n} D_{i}(J_{i}) = 0. On the other

hand, D_{t} = \frac{∂}{∂t} + �_{u_{t}}, therefore D_{t}(J_{0}) + ∑_{i=1}^{n} D_{i}(J_{i}) = \frac{∂J_{0}}{∂t} + �_{u_{t}}(J_{0}) −

\frac{∂J_{0}}{∂t} − �_{f}(J_{0}) = �_{u_{t}−f}(J_{0}) = ℓ_{J_{0}}(u_{t} − f). Thus ψ = ℓ_{J_{0}}^{∗}(1) = E(J_{0}).

Let φ ∈ ker ℓ_{E} be a symmetry and [ω] ∈ H̄^{n−1}(E) a conservation law of the equation E. Then [�_{φ}(ω)] is a conservation law of E as well.

Proposition 7.13. If ψ ∈ ker ℓ_{E}^{∗} is the generating function of a conservation law [ω] of an ℓ-normal equation E = {F = 0}, then the generating function of the conservation law [�_{φ}(ω)] has the form �_{φ}(ψ)+Δ^{∗}(ψ), where the operator Δ ∈ CDiff(P, P) is defined by �_{φ}(F) = Δ(F).

Proof. First, we have

⟨ψ, ℓ_{E}(χ)⟩ = d̄ℓ_{ω}(χ) + d̄γ(ℓ_{E}(χ)), χ ∈ κ,

where γ ∈ CDiff(P, Λ̄^{n−1}). Using the obvious formula

ℓ_{�_{χ_{1}}(η)} (χ_{2}) = �_{χ_{1}}(ℓ_{η}(χ_{2})) − ℓ_{η}({χ_{1}, χ_{2}}), χ_{1}, χ_{1} ∈ κ, η ∈ Λ̄^{n},

where {·, ·} is the Jacobi bracket (see Definition 3.31 on page 65, we obtain

d̄ℓ_{�_{φ}(ω)} (χ) = d̄(�_{φ}(ℓ_{ω}(χ)))−d̄(ℓ_{ω}({φ, χ})) = �_{φ}(d̄(ℓ_{ω}(χ)))−d̄(ℓ_{ω}({φ, χ}))

= �_{φ}(⟨ψ, ℓ_{E}(χ)⟩) − ⟨ψ, ℓ_{E}({φ, χ})⟩ − �_{φ}(d̄γ(ℓ_{E}(χ))) + d̄γ(ℓ_{E}({φ, χ}))

= ⟨�_{φ}(ψ), ℓ_{E}(χ)⟩ + ⟨ψ, (�_{φ}(ℓ_{E}(χ)) − ℓ_{E}({φ, χ}))⟩ + d̄γ′(ℓ_{E}(χ))

= ⟨�_{φ}(ψ), ℓ_{E}(χ)⟩ + ⟨ψ, ℓ_{�_{φ}(F)} |_{E^{∞}}⟩ + d̄γ′(ℓ_{E}(χ))
= ⟨�_{φ}(ψ), ℓ_{E}(χ)⟩ + ⟨ψ, Δ(ℓ_{E}(χ))⟩ + d̄γ′(ℓ_{E}(χ))

= ⟨(�_{φ} + Δ^{∗})(ψ), ℓ_{E}(χ)⟩ + d̄γ′′(ℓ_{E}(χ)),

where γ′, γ′′ ∈ CDiff(P, Λ̄^{n−1}). This completes the proof.

7.6. Generating functions from the antifield-BRST standpoint. A differential equation E = {F = 0}, F ∈ P, is called normal, if any C-differ-ential operator Δ, such that Δ(F) = 0, vanishes on E^{∞}. A normal equation is obviously ℓ-normal.

Consider a normal equation E and the complex on J^{∞}(π)

0 ⟵ F ⟵^{δ} CDiff(P, F) ⟵^{δ} CDiff_{(2)}^{alt}(P, F) ⟵^{δ} CDiff_{(3)}^{alt}(P, F) ⟵^{δ} ⋯,

δ(Δ)(p_{1}, …, p_{k}) = Δ(F, p_{1}, …, p_{k}), p_{i} ∈ P. This complex is exact in all terms except for the term F. At points θ ∈ E^{∞}, the exactness follows immediately from the normality condition. At points θ ∉ E^{∞}, this is a well known fact from linear algebra (see Example 8.5 on page 135). The homology in the term F is clearly equal to F(E).

In physics, this complex is said to be the Koszul–Tate resolution, and elements of CDiff_{(k)}^{alt}(P, F) are called antifields.

Consider the commutative diagram

0  0  0
↑  ↑  ↑
0 ⟵ Λ̄^{n} ⟵^{δ} CDiff(P, Λ̄^{n}) ⟵^{δ} CDiff_{(2)}^{alt}(P, Λ̄^{n}) ⟵^{δ} ⋯
↑d̄  ↑d̄  ↑d̄
0 ⟵ Λ̄^{n−1} ⟵^{δ} CDiff(P, Λ̄^{n−1}) ⟵^{δ} CDiff_{(2)}^{alt}(P, Λ̄^{n−1}) ⟵^{δ} ⋯
↑d̄  ↑d̄  ↑d̄
0 ⟵ Λ̄^{n−2} ⟵^{δ} CDiff(P, Λ̄^{n−2}) ⟵^{δ} CDiff_{(2)}^{alt}(P, Λ̄^{n−2}) ⟵^{δ} ⋯
↑d̄  ↑d̄  ↑d̄
⋮  ⋮  ⋮

From the standard spectral sequence arguments (see the Appendix) and Theorem 2.8 on page 32 it follows that H̄^{q}(E) = H_{n−q}(L_{•}^{alt}(P), δ). Since the complex (L_{•}^{alt}(P), δ) is a direct summand in the complex (CDiff_{(•)}^{alt}(P, P̂), δ), it is exact in all degrees except for 0 and 1. This yields the two-line theorem for normal equations. We also get

H̄^{n−1}(E) = H_{1}(L_{•}^{alt}(P), δ) = { ψ ∈ P̂ mod T | ⟨ψ, F ⟩ ∈ d̄Λ̄^{n−1} },

where T = { ψ ∈ P̂ | ψ = □(F), □ ∈ CDiff(P, P̂), □^{∗} = −□ }. The condition ⟨ψ, F ⟩ ∈ d̄Λ̄^{n−1} is equivalent to 0 = E⟨ψ, F ⟩ = ℓ_{F}^{∗} (ψ) + ℓ_{ψ}^{∗}(F). So we again obtain the correspondence between conservation laws and generating functions together with the equation ℓ_{E}^{∗}(ψ) = 0.

7.7. Euler–Lagrange equations. Consider the Euler–Lagrange equation E = {E(L) = 0} corresponding to a Lagrangian L = [ω] ∈ H̄^{n}(π). Let φ ∈ κ(π) be a Noether symmetry of L, i.e., �_{φ}(L) = 0 on J^{∞}(π).

Exercise 7.9. Using Exercise 7.7 on page 116, check that a Noether symmetry of L is a symmetry of the corresponding equation E as well, i.e., sym(L) ⊂ sym(E).

Exercise 7.10. Show that if E_{2}^{0,n}(E) = 0, then finding of Noether symmetries of the Lagrangian L = [ω] amounts to solution of the equation E(ℓ_{ω}(φ)) = ℓ_{E(L)}(φ) + ℓ_{φ}^{∗}(E(L)) = 0. (Thus, to calculate the Noether symmetries of an Euler–Lagrange equation one has no need to know the Lagrangian.)

Let �_{φ}(ω) = d̄ν, where ν ∈ Λ̄^{n−1}(π). By the Green formula we have

�_{φ}(ω) − d̄ν = ℓ_{ω}(φ) − d̄ν = ℓ_{ω}^{∗}(1)(φ) + d̄γ(φ) − d̄ν

= E(L)(φ) + d̄(γ(φ) − ν) = 0.

Set

η = (ν − γ(φ))|_{E^{∞}} ∈ Λ̄^{n−1}(E).

Thus, d̄η|_{E^{∞}} = 0, i.e., [η] ∈ H̄^{n−1}(E) is a conservation law of the equation E. The map

sym(L) → H̄^{n−1}(E), φ ↦ [η],

is said to be the Noether map.

An arbitrariness in the choice of ω and ν leads to the multivaluedness of the Noether map.

Exercise 7.11. Check that the Noether map is well defined up to the image of the natural homomorphism H̄^{n−1}(π) → H̄^{n−1}(E).

Proposition 7.14. If the Euler–Lagrange equation E corresponding to a Lagrangian L is ℓ-normal, then the Noether map considered on the set of Noether symmetries of L is inverse to the differential d_{1}^{0,n−1}.

Proof. On J^{∞}(π) we have

d̄ℓ_{η}(χ) = ℓ_{⟨E(L),φ⟩}(χ) = ⟨ℓ_{E(L)}(χ), φ⟩ + ⟨E(L), ℓ_{φ}(χ)⟩.

Therefore on E^{∞} we obtain d̄ℓ_{η}(χ) = ⟨ℓ_{E}(χ), φ⟩, i.e., d_{1}^{0,n−1}([η]) = φ.

Remark 7.5. The Noether map can be understood as a procedure for finding a conserved current corresponding to a given generating function.

Thus, we see that if �_{φ} is a Noether symmetry of a Lagrangian, then φ is the generating function of a conservation law for the corresponding Euler–Lagrange equation. This is the (first) Noether theorem. Note that since for Euler–Lagrange equations one has ℓ_{E}^{∗} = ℓ_{E}, the inverse Noether theorem is obvious: if φ is the generating function of a conservation law for an Euler–Lagrange equation, then φ is a symmetry for this equation.

Let us discuss the Noether theorem from the antifield-BRST point of view. Consider a 1-cycle φ ∈ κ of the complex L_{•}^{alt}(κ̂). We have ⟨φ, E(ω)⟩ ∈ d̄Λ̄^{n−1}, where ω is a density of the Lagrangian L = [ω]. Hence �_{φ}(ω) ∈ d̄Λ̄^{n−1} and, therefore, �_{φ}(L) = 0, i.e., φ is a Noether symmetry. Thus, the Koszul–Tate resolution gives a homological interpretation of the Noether theorem.

Now, suppose that the Lagrangian has a gauge symmetry, i.e., there exist an F-module 𝔞 and a C-differential operator R: 𝔞 → κ such that R(α) is a Noether symmetry for any α ∈ 𝔞. This means that �_{R(α)}(L) = 0 or ℓ_{L}∘R = 0. Hence R^{∗}∘ℓ_{L}^{∗} = 0 and, finally, R^{∗}(ℓ_{L}^{∗}(1)) = R^{∗}(E(L)) = 0. Thus, if the Lagrangian is invariant under a gauge symmetry, then theNoether identities R^{∗}(E(L)) = 0 between the Euler–Lagrange equations hold (the second Noether theorem).

7.8. The Hamiltonian formalism on J^{∞}(π). Let A ∈ CDiff(κ̂(π), κ(π)) be a C-differential operator. Define the Poisson bracket on H̄^{n}(π) corresponding to the operator A by the formula

{ω_{1}, ω_{2}}_{A} = ⟨A(E(ω_{1})), E(ω_{2})⟩,

where ⟨ , ⟩ denotes the natural pairing κ(π) × κ̂(π) → H̄^{n}(π).

The lemma below shows that the operator A is uniquely determined by the corresponding Poisson bracket.

Lemma 7.15. Let π: E → M be a vector bundle.

(1) Consider an operator A ∈ CDiff_{(l)}(κ̂(π), P), where P is an F(π)- module. If for all ω_{1}, …, ω_{l} ∈ H̄^{n}(π) one has

A(E(ω_{1}), …, E(ω_{l})) = 0,

then A = 0.

(2) Consider an operator A ∈ CDiff_{(l)}(κ̂(π), Λ̄^{n}(π)). If for all cohomology classes ω_{1}, …, ω_{l} ∈ H̄^{n}(π) the element A(E(ω_{1}), …, E(ω_{l})) belongs to the image of d̄, then im A ⊂ im d̄, i.e., μ_{(l−1)}(A) = 0 (see Subsection 2.4).

(3) Consider an operator A ∈ CDiff_{(l−1)}(κ̂(π), κ(π)). If for all elements ω_{1}, …, ω_{l} ∈ H̄^{n}(π) one has ⟨A(E(ω_{1}), …, E(ω_{l−1})), E(ω_{l})⟩ = 0, then A = 0.

Proof. (1) It suffices to consider the case l = 1. Obviously, on J^{∞}(π) every element of κ̂(π) = F̂(π, π) of the form π^{∗}(f), with f ∈ Γ̂(π), (in other words, every element of κ̂(π) depending on base coordinates x only) can locally be presented in the form π^{∗}(f) = E(ω) for some ω ∈ Λ̄^{n}(π). Thus A(π^{∗}(f)) = 0 for all f. Since A is a C-differential operator, this implies A = 0.

(2) It is also sufficient to consider the case l = 1. We have E(A(E(ω))) = 0. Using Exercise 7.7 on page 116, we get

0 = E(A(E(ω))) = ℓ_{E(ω)}^{∗}(A^{∗}(1)) + ℓ_{A^{∗}(1)}^{∗}(E(ω))

for all ω ∈ Λ̄^{n}(π). As above, we see that for any f ∈ Γ̂(π) there exists ω ∈ Λ̄^{n}(π) such that π^{∗}(f) = E(ω). Since ℓ_{π^{∗}(f)} = 0, we obtain ℓ_{A^{∗}(1)}^{∗}(π^{∗}(f)) = 0. Hence ℓ_{A^{∗}(1)}^{∗} = 0, so that 0 = E(A(E(ω))) = ℓ_{E(ω)}^{∗}(A^{∗}(1)).

Exercise 7.12. Prove that locally there exists a form ω ∈ Λ̄^{n}(π) such that ℓ_{E(ω)} is the identity operator.

Using this exercise, we get 0 = A^{∗}(1) = μ(A), which is our claim. (3) The assertion follows immediately from (1) and (2) above.

Definition 7.1. An operator A ∈ CDiff(κ̂(π), κ(π)) is called Hamiltonian, if its Poisson bracket defines a Lie algebra structure on H̄^{n}(π), i.e., if

{ω_{1}, ω_{2}}_{A} = −{ω_{2}, ω_{1}}_{A},  (7.4)
{{ω_{1}, ω_{2}}_{A}, ω_{3}}_{A} + {{ω_{2}, ω_{3}}_{A}, ω_{1}}_{A} + {{ω_{3}, ω_{1}}_{A}, ω_{2}}_{A} = 0.  (7.5)

The bracket { , }_{A} is said to be a Hamiltonian structure.

Proposition 7.16. The Poisson bracket { , }_{A} is skew-symmetric, i.e., condition (7.4) holds, if and only if the operator A is skew-adjoint, i.e., A = −A^{∗}.

Proof. Since

{ω_{1}, ω_{2}}_{A} + {ω_{2}, ω_{1}}_{A} = ⟨(A + A^{∗})(E(ω_{1})), E(ω_{2})⟩,

the claim follows immediately from the previous lemma.

Now we shall prove criteria for checking an arbitrary skew-adjoint operator A ∈ CDiff(κ̂(π), κ(π)) to be Hamiltonian. For this, we need the following

Lemma 7.17. Consider an operator A ∈ CDiff(κ̂(π), κ(π)) and an element ψ ∈ κ̂(π). Define the operator ℓ_{A,ψ} ∈ CDiff(κ(π), κ(π)) by

ℓ_{A,ψ}(φ) = (ℓ_{A}(φ))(ψ) φ ∈ κ(π).

Then

ℓ_{A,ψ_{1}}^{∗}(ψ_{2}) = ℓ_{A^{∗},ψ_{2}}^{∗}(ψ_{1}).  (7.6)

Proof. By the Green formula,

⟨A(ψ_{1}), ψ_{2}⟩ = ⟨ψ_{1}, A^{∗}(ψ_{2})⟩.

Applying �_{φ} to both sides, we get

⟨�_{φ}(A)(ψ_{1}), ψ_{2}⟩ = ⟨ψ_{1}, �_{φ}(A^{∗})(ψ_{2})⟩,

and so

⟨ℓ_{A,ψ_{1}}(φ), ψ_{2}⟩ = ⟨ψ_{1}, ℓ_{A^{∗},ψ_{2}}(φ)⟩.

Again the Green formula yields

⟨φ, ℓ_{A,ψ_{1}}^{∗}(ψ_{2})⟩ = ⟨ℓ_{A^{∗},ψ_{2}}^{∗}(ψ_{1}), φ⟩,

and the lemma is proved.

Theorem 7.18. Let A ∈ CDiff(κ̂(π), κ(π)) be a skew-adjoint operator; then the following conditions are equivalent:

(1) A is a Hamiltonian operator;

(2) ⟨ℓ_{A}(A(ψ_{1}))(ψ_{2}), ψ_{3}⟩ + ⟨ℓ_{A}(A(ψ_{2}))(ψ_{3}), ψ_{1}⟩ + ⟨ℓ_{A}(A(ψ_{3}))(ψ_{1}), ψ_{2}⟩ = 0 for all ψ_{1}, ψ_{2}, ψ_{3} ∈ κ̂(π);

(3) ℓ_{A,ψ_{1}}(A(ψ_{2})) − ℓ_{A,ψ_{2}}(A(ψ_{1})) = A(ℓ_{A,ψ_{2}}^{∗}(ψ_{1})) for all ψ_{1}, ψ_{2} ∈ κ̂(π); (4) the expression ℓ_{A,ψ_{1}}(A(ψ_{2}))+ \frac{1}{2}A(ℓ_{A,ψ_{1}}^{∗}(ψ_{2})) is symmetric with respect

to ψ_{1}, ψ_{2} ∈ κ̂(π);

(5) [�_{A(ψ)}, A] = ℓ_{A(ψ)} ∘ A + A ∘ ℓ_{A(ψ)}^{∗} for all ψ ∈ im E ⊂ κ̂(π). Moreover, it is sufficient to verify conditions (2)–(4) for elements ψ_{i} ∈ im E only.

Proof. Let ω_{1}, ω_{2}, ω_{3} ∈ H̄^{n}(π) and ψ_{i} = E(ω_{i}). The Jacobi identity (7.5) on the page before yields

∮ {{ω_{1}, ω_{2}}_{A}, ω_{3}}_{A} = ∮ −�_{A(ψ_{3})}⟨A(ψ_{1}), ψ_{2}⟩

= ∮ −⟨�_{A(ψ_{3})}(A)(ψ_{1}), ψ_{2}⟩ − ⟨A(ℓ_{ψ_{1}}(A(ψ_{3}))), ψ_{2}⟩ − ⟨A(ψ_{1}), ℓ_{ψ_{2}}(A(ψ_{3}))⟩

= ∮ −⟨ℓ_{A}(A(ψ_{3}))(ψ_{1}), ψ_{2}⟩ + ⟨A(ψ_{2}), ℓ_{ψ_{1}}(A(ψ_{3}))⟩ − ⟨A(ψ_{1}), ℓ_{ψ_{2}}(A(ψ_{3}))⟩

= ∮ −⟨ℓ_{A}(A(ψ_{3}))(ψ_{1}), ψ_{2}⟩ = 0,

where as above the symbol ∮ denotes the sum of cyclic permutations. It follows from Lemma 7.15 on page 126 that this formula holds for all ψ_{i} ∈ κ̂(π). Criterion (2) is proved.

Rewrite the Jacobi identity in the form

⟨ℓ_{A,ψ_{1}}(A(ψ_{2})), ψ_{3}⟩ + ⟨A(ψ_{1}), ℓ_{A,ψ_{3}}^{∗}(ψ_{2})⟩ − ⟨A(ℓ_{A,ψ_{2}}^{∗}(ψ_{1})), ψ_{3}⟩ = 0.

Using (7.6) on the page before, we obtain

⟨ℓ_{A,ψ_{1}}(A(ψ_{2})), ψ_{3}⟩ − ⟨ℓ_{A,ψ_{2}}(A(ψ_{1})), ψ_{3}⟩ − ⟨A(ℓ_{A,ψ_{2}}^{∗}(ψ_{1})), ψ_{3}⟩ = 0,

which implies criterion (3).

The equivalence of criteria (3) and (4) follows from (7.6) on the preceding page.

Finally, criterion (5) is equivalent to criterion (3) by virtue of the following obvious equalities:

[�_{A(ψ_{2})}, A](ψ_{1}) = ℓ_{A,ψ_{1}}(A(ψ_{2})),
ℓ_{A,ψ} ∘ A = ℓ_{A(ψ)} ∘ A − A ∘ ℓ_{ψ} ∘ A.

This concludes the proof.

Example 7.2. Consider a skew-symmetric differential operator Δ: Γ̂(π) → Γ(π). Then its lifting (see Definition 3.25 on page 58) CΔ: κ̂(π) → κ(π) is obviously a Hamiltonian operator.

Exercise 7.13. Check that in the case n = dim M = 1 and m = dim π = 1 operators of the form A = D_{x}^{3} + (α + βu)D_{x} + \frac{β}{2} u_{x} are Hamiltonian.

Let A: κ̂(π) → κ(π) be a Hamiltonian operator. For any ω ∈ H̄^{n}(π) the evolutionary vector field X_{ω} = �_{A(E(ω))} is called Hamiltonian vector field corresponding to the Hamiltonian ω. Obviously,

X_{ω_{1}}(ω_{2}) = ⟨AE(ω_{1}), E(ω_{2})⟩ = {ω_{1}, ω_{2}}_{A}.

This yields

X_{{ω_{1},ω_{2}}_{A}}(ω) = {{ω_{1}, ω_{2}}_{A}, ω}_{A} = {ω_{1}, {ω_{2}, ω}_{A}}_{A} − {ω_{2}, {ω_{1}, ω}_{A}}_{A} = (X_{ω_{1}} ∘ X_{ω_{2}} − X_{ω_{2}} ∘ X_{ω_{1}})(ω) = [X_{ω_{1}}, X_{ω_{2}}](ω)

for all ω ∈ H̄^{n}(π). Thus

X_{{ω_{1},ω_{2}}_{A}} = [X_{ω_{1}}, X_{ω_{2}}].  (7.7)

As with the finite dimensional Hamiltonian formalism, 7.7 implies a result similar to the Noether theorem.

For each H ∈ H̄^{n}(π), the evolution equation

u_{t} = A(E(H)),  (7.8)

corresponding to the Hamiltonian H is called Hamiltonian evolution equation.

Example 7.3. The KdV equation u_{t} = uu_{x}+u_{xxx} admits two Hamiltonian structures:

u_{t} = D_{x} (E (\frac{u^{3}}{6} − \frac{u_{x}^{2}}{2} ))

and

u_{t} = (D_{x}^{3} + \frac{2}{3}uD_{x} + \frac{1}{3}u_{x}) (E (\frac{u^{2}}{2} )) .

Theorem 7.19. Hamiltonian operators take the generating function of a conservation law of equation (7.8) to the symmetry of this equation.

Proof. Let A be a Hamiltonian operator and

ω̃_{0}(t) + ω̃_{1}(t) ∧ dt ∈ Λ̄^{n}(π) ⊕ Λ̄^{n−1}(π) ∧ dt

be a conserved current of equation (7.8). This means that D_{t}(ω_{0}(t)) = 0, where ω_{0}(t) ∈ H̄^{n}(π) is the horizontal cohomology class corresponding to the form ω̃_{0}(t), and D_{t} is the restriction of the total derivative in t to the equation. Further,

D_{t}(ω_{0}) = \frac{∂ω_{0}}{∂t} + �_{A(E(H))}(ω_{0}) = \frac{∂ω_{0}}{∂t} + \{H, ω_{0}\}.

This yields

\frac{∂}{∂t}X_{ω_{0}} + [X_{H}, X_{ω_{0}}] = 0.

Hence X_{ω_{0}} = �_{A(E(ω_{0}))} is a symmetry of (7.8) on the page before. It remains to recall that E(ω_{0}) is the generating function of the conservation law under consideration (see Proposition 7.12 on page 122).

Remark 7.6. Thus Hamiltonian operators are in a sense dual to elements of E_{1}^{2,n−1} (cf. Remark 7.4 on page 122).

7.9. On superequations. The theory of this and preceding sections is based on the pure algebraic considerations in Sections 1 and 2. Therefore all results remain valid for the case of differential superequations, provided one inserts the minus sign where appropriate (detailed geometric definitions of superjets, super Cartan distribution, and so on the reader can find, for example, in [44, 45]). So we discuss here only a couple of somewhat less obvious points and the coordinates formula.

Let M be a supermanifold, dim M = n|m, and π be a superbundle over M, dim π = s|t. The following theorem is the superanalog of theorem 2.2 on page 28.

Theorem 7.20. (1) Â_{s} = 0 for s ≠ n.

(2) Â_{n} is the module of sections for the bundle Ber(M), the latter being defined as follows: locally, sections of Ber(M) are written in the form f(x)D(x), where f ∈ C^{∞}(U) and D is a basis local section that is multiplied by the Berezin determinant of the Jacobi matrix under the change of coordinates. The Berezin determinant of an even matrix ( \begin{matrix}A & B \\ C & D \end{matrix} ) is equal to det(A − BD^{−1}C)(det D)^{−1}.

Proof. The assertion is local, so we can consider the domain U with local coordinates x = (y_{i}, ξ_{j}), i = 1, …, n, j = 1, …, m, and split the complex (2.1) on page 27 Diff^{+}(Λ^{∗}) in the tensor product of complexes Diff^{+}(Λ^{∗})_{even} ⊗ Diff^{+}(Λ^{∗})_{odd}, where Diff^{+}(Λ^{∗})_{even} is complex (2.1) on the underlying even domain of U and Diff^{+}(Λ^{∗})_{odd} is the same complex for the Grassmann algebra in variables ξ_{1}, …, ξ_{m}.

We have H^{i}(Diff^{+}(Λ^{∗})_{even}) = 0 for i ≠ n and H^{i}(Diff^{+}(Λ^{∗})_{even}) = Λ_{U}^{n}, where Λ_{U}^{n} is the module of n-form on the underlying even domain of U. To compute the cohomology of Diff^{+}(Λ^{∗})_{odd} consider the quotient complexes

0 ⟶ Smbl_{k}(A)_{odd} ⟶ Smbl_{k+1}(Λ^{1})_{odd} ⟶ ⋯,

where Smbl_{k}(P)_{odd} = Diff_{k}^{+}(P)_{odd}/Diff_{k−1}^{+}(P)_{odd}. Then an easy calculation shows that these complexes are the Koszul complexes, hence H^{i}(Diff^{+}(Λ^{∗}))_{odd} = 0 for i > 0 and H^{0}(Diff^{+}(Λ^{∗})) is a module of rank 1. Therefore Â_{i} = H^{i}(Diff^{+}(Λ^{∗})) = 0 for i ≠ n and the only operators that

represent non-trivial cocycles have the form dy_{1}∧⋯∧dy_{n} \frac{∂^{m}}{∂ξ_{1} ⋯ ∂ξ_{m}} f(y, ξ).

To complete the proof it remains to check that Â_{n} is precisely Ber(M), i.e., that changing coordinates we obtain:

dy_{1} ∧ ⋯ ∧ dy_{n} \frac{∂^{m}}{∂ξ_{1} … ∂ξ_{m}} f

= dv_{1} ∧ ⋯ ∧ dv_{n} \frac{∂^{m}}{∂η_{1} … ∂η_{m}} fBer (J (\frac{x}{z})) + T,

where z = (v_{i}, η_{j}) is a new coordinate system on U, Ber denotes the Berezin determinant, J (\frac{x}{z}) is the Jacobi matrix, T is cohomologous to zero. This is an immediate consequence of the following well known formula for the Berezin determinant: Ber( \begin{matrix}A & B \\ C & D \end{matrix} ) = det A · det D̃, where D̃ is defined by

( \begin{matrix}A & B \\ C & D \end{matrix} )^{−1} = ( \begin{matrix}Ã & B̃ \\ C̃ & D̃ \end{matrix} ).

The coordinate expression for the adjoint operator is as follows. Let

Δ ∈ Diff(A, B) be a scalar operator Δ = ∑_{σ} Da_{σ} \frac{∂^{|σ|}}{∂x_{σ}}. Then

Δ^{∗} = ∑_{σ}(−1)^{|σ|+a_{σ}x_{σ}}D\frac{∂^{|σ|}}{∂x_{σ}} ∘ a_{σ}.

Here the symbol of an object used in exponent denotes the parity of the object.

Now, consider a matrix operator Δ: P → Q, Δ = ∥Δ_{j}^{i}∥, where the matrix elements are defined by the equalities Δ(∑_{α} e_{α}f^{α}) = ∑_{α,β} e′_{α}Δ_{β}^{α}(f^{β}), {e_{i}} is a basis in P, {e′_{i}} is a basis in Q. If D is even, then Δ^{∗} has the form

D(Δ^{∗})_{j}^{i} = (−1)^{(e_{i}+e′_{j})(Δ+e_{i})}(DΔ_{i}^{j})^{∗}.

If D is odd, then

D((Δ^{∗})^{Π})_{j}^{i} = (−1)^{(e_{i}+Δ)(e′_{j}+1)+Δe_{i}}(DΔ_{i}^{j})^{∗},

where ( \begin{matrix}A & B \\ C & D \end{matrix} )^{Π} = ( \begin{matrix}D & C \\ B & A \end{matrix} )is the Π-transposition.

Remark 7.7. One has (Δ^{∗∗})_{j}^{i} = (−1)^{e_{i}+e′_{j}}Δ_{j}^{i}.

Remark 7.8. There is one point where we need to improve the algebraic theory of differential operators to extend it to the supercase. This is the definition of geometrical modules that should read:

Definition 7.2. A module P over C^{∞}(M) is called geometrical, if

⋂_{\begin{matrix}x ∈ M_{rd} \\ k ≥ 1 \end{matrix}} μ_{x}^{k}P = 0,

where M_{rd} is the underlying even manifold of M and μ_{x} is the ideal in C^{∞}(M) consisting of functions vanishing at point x ∈ M_{rd}.

Appendix: Homological algebra

In this appendix we sketch the basics of homological algebra. For an extended discussion see, e.g., [37, 20, 7, 41, 8].

8.1. Complexes. A sequence of vector spaces over a field 𝕜 and linear mappings

⋯ ⟶ K^{i−1} ⟶^{d^{i−1}} K^{i} ⟶^{d^{i}} K^{i+1} ⟶^{d^{i+1}} ⋯

is said to be a complex if the composition of any two neighboring arrows is the zero map: d^{i} ∘ d^{i−1} = 0.

The maps d^{i} are called differentials. The index i is often omitted, so that the definition of a complex reads: d^{2} = 0.

By definition, im d^{i−1} ⊂ ker d^{i}. The complex (K^{•}, d^{•}) is called exact (or acyclic) in degree i, if im d^{i−1} = ker d^{i}. A complex exact in all degrees is called acyclic (or exact, or an exact sequence).

Example 8.1. The sequence 0 ⟶ L ⟶^{f} K is always a complex. It is acyclic if and only if f is injection. The sequence K ⟶^{g} M ⟶ 0 is always a complex, as well. It is acyclic if and only if g is surjection.

The sequence

0 ⟶ L ⟶^{f} K ⟶^{g} M ⟶ 0  (8.1)

is a complex, if g ∘ f = 0. It is exact, if and only if f is injection, g is surjection, and im f = ker g. In this case we can identify L with a subspace of K and M with the quotient space K/L. Exact sequence (8.1) is called a short exact sequence (or an exact triple).

Example 8.2. The de Rham complex is the complex of differential forms on a smooth manifold M with respect to the exterior derivation:

⋯ ⟶ Λ^{i−1} ⟶^{d} Λ^{i} ⟶^{d} Λ^{i+1} ⟶^{d} · · · .

The cohomology of a complex (K^{•}, d^{•}) is the family of the spaces

H^{i}(K^{•}, d^{•}) = ker d^{i}/ im d^{i−1}.

Thus, the equality H^{i}(K^{•}, d^{•}) = 0 means that the complex (K^{•}, d^{•}) is acyclic in degree i. Note that for the sake of brevity the cohomology is often denoted by H^{i}(K^{•}) or H^{i}(d^{•}). Elements of ker d^{i} ⊂ K^{i} are called i-dimensional cocycles, elements of im d^{i−1} ⊂ K^{i} are called i-dimensional coboundaries. Thus, the cohomology is the quotient space of the space of all cocycles by the subspace of all coboundaries. Two cocycles k_{1} and k_{2} from common cohomology coset, i.e., such that k_{1} − k_{2} ∈ im d^{i−1}, are called cohomologous.

Remark 8.1. In the case of the complex of differential forms on a manifold cocycles are called closed forms, and coboundaries are called exact forms.

Remark 8.2. It is clear that the definition of a complex can be immediately generalized to modules over a ring instead of vector spaces.

Exercise 8.1. Prove that if

⋯ ⟶ Q^{i−1} ⟶^{d^{i−1}} Q^{i} ⟶^{d^{i}} Q^{i+1} ⟶^{d^{i+1}} ⋯

is a complex of modules (and d^{i} are homomorphisms) and P is a projective module, then H^{i}(Q^{•} ⊗ P) = H^{i}(Q^{•}) ⊗ P.

Complexes defined above are called cochain to stress that the differentials raise the dimension by 1. Inversion of arrows gives chain complexes

⋯ ⟵^{d_{i−1}} K_{i−1} ⟵^{d_{i}} K_{i} ⟵^{d_{i+1}} K_{i+1} ⟵ ⋯,

homology, cycles, boundaries, etc. The difference between these types of complex is pure terminological, so we shall mainly restrict our considerations to cochain complexes.

A morphism (or a cochain map) of complexes f: K^{•} → L^{•} is the family of linear mappings f^{i}: K^{i} → L^{i} that commute with differentials, i.e., that make the following diagram commutative:

⋯ ⟶ K^{i−1} ⟶^{d_{K}^{i−1}} K^{i} ⟶^{d_{K}^{i}} K^{i+1} ⟶^{d_{K}^{i+1}} ⋯
↓^{f^{i−1}}  ↓^{f^{i}}  ↓^{f^{i+1}}
⋯ ⟶ L^{i−1} ⟶^{d_{L}^{i−1}} L^{i} ⟶^{d_{L}^{i}} L^{i+1} ⟶^{d_{L}^{i+1}} · · · .

Such a morphism induces the map H^{i}(f): H^{i}(K^{•}) → H^{i}(L^{•}), [k] ↦ [f(k)], where k is a cocycle and [ · ] denotes the cohomology coset. Clearly, H^{i}(f ∘ g) = H^{i}(f) ∘ H^{i}(g) (so that H^{i} is a functor from the category of complexes to the category of vector spaces). A morphism of complexes is called quasiisomorphism (or homologism) if it induces an isomorphism of cohomologies.

Example 8.3. A smooth map of manifolds F: M_{1} → M_{2} gives rise to the map of differential forms F^{∗}: Λ^{•}(M_{2}) → Λ^{•}(M_{1}), such that d(F^{∗}(ω)) = F^{∗}(d(ω)). Thus F^{∗} is a cochain map and induces the map of the de Rham cohomologies F^{∗}: H^{•}(M_{2}) → H^{•}(M_{1}). In particular, if M_{1} and M_{2} are diffeomorphic, then their de Rham cohomologies are isomorphic.

Exercise 8.2. Check that the wedge product on differential forms on M induces a well-defined multiplication on the de Rham cohomology H^{∗}(M) = ⨁_{i} H^{i}(M), which makes the de Rham cohomology a (super)algebra, and not just a vector space. Show that for diffeomorphic manifolds these algebras are isomorphic.

Two morphisms of complexes f^{•}, g^{•}: K^{•} → L^{•} are called homotopic if there exist mappings s^{i}: K^{i} → L^{i−1}, such that

f^{i} − g^{i} = s^{i+1}d^{i} + d^{i−1}s^{i}.

The mappings s^{i} are called (cochain) homotopy.

Proposition 8.1. If morphisms f^{•} and g^{•} are homotopic, then H^{i}(f^{•}) = H^{i}(g^{•}) for all i.

Proof. Consider a cocycle z ∈ K^{i}, dz = 0. Then

f(z) − g(z) = (sd + ds)(z) = d(s(z)).

Thus, f(z) and g(z) are cohomologous, and so H^{i}(f^{•}) = H^{i}(g^{•}).

Two complexes K^{•} and L^{•} are said to be cochain equivalent if there exist morphisms f^{•}: K^{•} → L^{•} and g^{•}: L^{•} → K^{•} such that g ∘ f is homotopic to id_{K^{•}} and f ∘g is homotopic to id_{L^{•}}. Obviously, cochain equivalent complexes have isomorphic cohomologies.

Example 8.4. Consider two maps of smooth manifolds F_{0}, F_{1} : M_{1} → M_{2} and assume that they are homotopic (in the topological sense). Let us show that the corresponding morphisms of the de Rham complexes F_{0}^{∗}, F_{1}^{∗}: Λ^{•}(M_{2}) → Λ^{•}(M_{1}) are homotopic (in the above algebraic sense).

Let F: M_{1} × [0, 1] → M_{2} be the homotopy between F_{0} and F_{1}, F_{0}(x) = F (x, 0), F_{1}(x) = F (x, 1). Take a form ω ∈ Λ^{i}(M_{2}). Then

F^{∗}(ω) = ω_{1}(t) + dt ∧ ω_{2}(t),

where ω_{1}(t) ∈ Λ^{i}(M_{1}), ω_{2}(t) ∈ Λ^{i−1}(M_{1}) for each t ∈ [0, 1]. In particular, F_{0}^{∗}(ω) = ω_{1}(0) and F_{1}^{∗}(ω) = ω_{1}(1). Set s(ω) = ∫_{0}^{1} ω_{2}(t) dt. We have F^{∗}(dω) = d(F^{∗}(ω)) = dω_{1}(t) + dt ∧ ω′_{1}(t) − dt ∧ dω_{2}(t), where ′ denotes the derivative in t. Hence, s(d(ω)) = ∫_{0}^{1}(ω′_{1}(t) − dω_{2}(t)) dt = ω_{1}(1) − ω_{1}(0) − d ∫_{0}^{1} ω_{2}(t) dt = F_{1}^{∗}(ω) − F_{0}^{∗}(ω) − d(s(ω)), so s is a homotopy between F_{0}^{∗} and F_{1}^{∗}.

Exercise 8.3. Prove that if two manifolds M_{1} and M_{2} are homotopic (i.e., there exist maps f: M_{1} → M_{2} and g: M_{2} → M_{1} such that the maps f ∘ g and g ∘ f are homotopic to the identity maps), then their cohomology are isomorphic.

Corollary 8.2 (Poincaré lemma). Locally, every closed form ω ∈ Λ^{i}(M), dω = 0, i ≥ 1, is exact: ω = dη.

A complex K^{•} is said to be homotopic to zero if the identity morphism id_{K^{•}} homotopic to the zero morphism, i.e., if there exist maps s^{i}: K^{i} → K^{i−1} such that id_{K^{•}} = sd + ds. Obviously, a complex homotopic to zero has the trivial cohomology.

Example 8.5. Let V be a vector space. Take a nontrivial linear functional u: V → 𝕜 and consider the complex

0 ⟵ 𝕜 ⟵^{d} V ⟵^{d} Λ^{2}(V) ⟵^{d} ⋯ ⟵^{d} Λ^{n−1}(V) ⟵^{d} Λ^{n}(V) ⟵^{d} ⋯,

where d is the inner product with u:

d(v_{1} ∧ ⋯ ∧ v_{k}) = ∑_{i=1}^{k}(−1)^{i+1}u(v_{i})v_{1} ∧ ⋯ ∧ v_{i−1} ∧ v_{i+1} ∧ ⋯ ∧ v_{k}.

Take also a nontrivial element v ∈ V and consider the complex

0 ⟶ 𝕜 ⟶^{s} V ⟶^{s} Λ^{2}(V) ⟶^{s} ⋯ ⟶^{s} Λ^{n−1}(V) ⟶^{s} Λ^{n}(V) ⟶^{s} ⋯,

where s is the exterior product with v:

s(v_{1} ∧ ⋯ ∧ v_{k}) = v ∧ v_{1} ∧ ⋯ ∧ v_{k}.

Since d is a derivation of the exterior algebra Λ^{∗}(V), we have (ds+sd)(w) = d(v ∧w)+v ∧dw = dv ∧w = u(v)w. This means that both complexes under consideration are homotopic to zero and, therefore, acyclic.

Example 8.6. Consider two complexes

0 ⟵ S^{n}(V) ⟵^{d} S^{n−1}(V) ⊗ V ⟵^{d} S^{n−2}(V) ⊗ Λ^{2}(V) ⟵^{d} ⋯,  (8.2)
0 ⟶ S^{n}(V) ⟶^{s} S^{n−1}(V) ⊗ V ⟶^{s} S^{n−2}(V) ⊗ Λ^{2}(V) ⟶^{s} ⋯,  (8.3)

where

d(w ⊗ v_{1} ∧ ⋯ ∧ v_{q}) = ∑_{i=1}^{q}(−1)^{i+1}v_{i}w ⊗ v_{1} ∧ ⋯ ∧ v_{i−1} ∧ v_{i+1} ∧ ⋯ ∧ v_{q},

s(w_{1} ⋯ w_{p} ⊗ v) = ∑_{i=1}^{p} w_{1} ⋯ w_{i−1}w_{i+1} ⋯ w_{p} ⊗ w_{i} ∧ v.

Both maps d and s are derivations of the algebra S^{∗}(V) ⊗ Λ^{∗}(V), therefore their commutator is also a derivation. Noting that on elements of degree one the commutator is identical, we get the formula

(ds + sd)(x) = (p + q)x, x ∈ S^{p}(V) ⊗ Λ^{q}(V).

Thus again both complexes under consideration are homotopic to zero (for n > 0). Complex (8.2) is called the Koszul complex. Complex (8.3) is the polynomial de Rham complex.

A complex L^{•} is called a subcomplex of a complex K^{•}, if the spaces L^{i} are subspaces of K^{i}, and the differentials of L^{•} are restrictions of differentials of K^{•}, i.e., d_{K}(L^{i−1}) ⊂ L^{i}. In this situation, differentials of K^{•} induce differentials on quotient spaces M^{i} = K^{i}/L^{i} and we obtain the complex M^{•} called the quotient complex and denoted by M^{•} = K^{•}/L^{•}.

The cohomologies of complexes K^{•}, L^{•}, and M^{•} = K^{•}/L^{•} are related to one another by the following important mappings. First, the inclusion φ: L^{•} → K^{•} and the natural projection ψ: K^{•} → M^{•} induce the cohomology mappings H^{i}(φ): H^{i}(L^{•}) → H^{i}(K^{•}) and H^{i}(ψ): H^{i}(K^{•}) → H^{i}(M^{•}). There exists one more somewhat less obvious mapping

∂^{i}: H^{i}(M^{•}) → H^{i+1}(L^{•})

called the boundary (or connecting) mapping.

The map ∂^{i} is defined as follows. Consider a cohomology class x ∈ H^{i}(M^{•}) represented by an element y ∈ M^{i}. Take an element z ∈ K^{i} such that ψ(z) = y. We have ψ(dz) = dψ(z) = dy = 0, hence there exists an element w ∈ L^{i+1} such that φ(w) = dz. Since φ(dw) = dφ(w) = ddz = 0, we get dw = 0, i.e., w is a cocycle. It can easily be checked that its cohomology class is independent of the choice of y and z. This class is the class ∂^{i}(x).

Thus, given a short exact sequence of complexes

0 ⟶ L^{•} ⟶^{φ} K^{•} ⟶^{ψ} M^{•} ⟶ 0  (8.4)

(this means that φ and ψ are morphisms of complexes and for each i the

sequences 0 ⟶ L^{i} ⟶^{φ^{i}} K^{i} ⟶^{ψ^{i}} M^{i} ⟶ 0 are exact), one has the following infinite sequence:

⋯ ⟶^{H^{i−1}(ψ)} H^{i−1}(M^{•}) ⟶^{∂^{i−1}} H^{i}(L^{•}) ⟶^{H^{i}(φ)} H^{i}(K^{•}) ⟶^{H^{i}(ψ)} H^{i}(M^{•})

⟶^{∂^{i}} H^{i+1}(L^{•}) ⟶^{H^{i+1}(φ)} ⋯ (8.5)

The main property of this sequence is the following.

Theorem 8.3. Sequence (8.5) is exact.

Proof. The proof is straightforward and is left to the reader.

Sequence (8.5) is called the long exact sequence corresponding to short exact sequence of complexes (8.4).

Exercise 8.4. Consider the commutative diagram

0 ⟶ A_{1} ⟶ A_{2} ⟶ A_{3} ⟶ 0
↓^{f}  ↓^{g}  ↓^{h}
0 ⟶ B_{1} ⟶ B_{2} ⟶ B_{3} ⟶ 0.

Prove using Theorem 8.3 that if f and h are isomorphisms, then g is also an isomorphism.

8.2. Spectral sequences. Given a complex K^{•} and a subcomplex L^{•} ⊂ K^{•}, the exact sequence (8.5) on the preceding page can tell something about the cohomology of K^{•}, if the cohomology of L^{•} and K^{•}/L^{•} are known. Now, suppose that we are given a filtration of K^{•}, that is a decreasing sequence of subcomplexes

K^{•} ⊃ K_{1}^{•} ⊃ K_{2}^{•} ⊃ K_{3}^{•} ⊃ · · · .

Then we obtain for each p = 0, 1, 2, … complexes

⋯ ⟶ E_{0}^{p,q−1} ⟶ E_{0}^{p,q} ⟶ E_{0}^{p,q+1} ⟶ ⋯,

where E_{0}^{p,q} = K_{p}^{p+q}/K_{p+1}^{p+q}. The cohomologies E_{1}^{p,q} = H^{p+q}(E_{0}^{p,•}) of these complexes can be considered as the first approximation to the cohomology of K^{•}. The apparatus of spectral sequences enables one to construct all successive approximations E_{r}, r ≥ 1.

Definition 8.1. A spectral sequence is a sequence of vector spaces E_{r}^{p,q}, r ≥ 0, and linear mappings d_{r}^{p,q}: E_{r}^{p,q} → E_{r}^{p+r,q−r+1}, such that d_{r}^{2} = 0 (more precisely, d_{r}^{p+r,q−r+1} ∘ d_{r}^{p,q} = 0) and the cohomology H^{p,q}(E_{r}^{•,•}, d_{r}^{•,•}) with respect to the differential d_{r} is isomorphic to E_{r+1}^{p,q}.

Thus E_{r} and d_{r} determine E_{r+1}, but do not determine d_{r+1}.

Usually, p + q, p, and q are called respectively the degree, the filtration degree, and the complementary degree.

It is convenient for each r to picture the spaces E_{r}^{p,q} as integer points on the plain with coordinates (p, q). The action of the differential d_{r} is shown as follows:

[figure: p; q; (p, q); (p + r, q − r + 1); E_{r}]

Take an element α ∈ E_{r}^{p,q}. If d_{r}(α) = 0 then α can be considered as an element of E_{r+1}^{p,q}. If again d_{r+1}(α) = 0 then α can be considered as an element of E_{r+2}^{p,q} and so on. This allows us to define the following two vector spaces:

C_{∞}^{p,q} = { α ∈ E_{0}^{p,q} | d_{0}(α) = 0, d_{1}(α) = 0, …, d_{r}(α) = 0, … },  (8.6)
B_{∞}^{p,q} = { α ∈ C_{∞}^{p,q} | there exists an element β ∈ E_{r}^{p,q} such that α = d_{r}(β) }.

Set E_{∞}^{p,q} = C_{∞}^{p,q}/B_{∞}^{p,q}. A spectral sequence is called regular if for any p and q there exists r_{0}, such that d_{r}^{p,q} = 0 for r ≥ r_{0}. In this case there are natural projections

E_{r}^{p,q} ⟶ E_{r+1}^{p,q} ⟶ ⋯ ⟶ E_{∞}^{p,q}, r ≥ r_{0},

and E_{∞}^{p,q} = inj lim E_{r}^{p,q}.

Let E and ′E be two spectral sequences. A morphism f: E → ′E is a family of mappings f_{r}^{p,q}: E_{r}^{p,q} → ′E_{r}^{p,q}, such that d_{r} ∘ f_{r} = f_{r} ∘ d_{r} and f_{r+1} = H(f_{r}). Obviously, a morphism f: E → ′E induces the maps f_{∞}^{p,q}: E_{∞}^{p,q} → ′E_{∞}^{p,q}. Further, it is clear that if f_{r} is an isomorphism, then f_{s} are isomorphisms for all s ≥ r. Moreover, if the spectral sequences E and ′E are regular, then f_{∞} is an isomorphism as well.

Exercise 8.5. Assume that E_{r}^{p,q} ≠ 0 for p ≥ p_{0}, q ≥ q_{0} only. Prove that in this case there exists r_{0} such that E_{r}^{p,q} = E_{r+1}^{p,q} = ⋯ = E_{∞}^{p,q} for r ≥ r_{0}.

Consider a graded vector space G = ⨁_{i∈ℤ} G^{i} endowed with a decreasing filtration ⋯ ⊃ G_{p} ⊃ G_{p+1} ⊃ ⋯, such that z ⋂_{p} G_{p} = 0 and ⋃_{p} G_{p} = G. The filtration is called regular, if for each i there exists p, such that G_{p}^{i} = 0.

It is said that a spectral sequence E converges to G, if the spectral sequence and the filtration of G are regular and E_{∞}^{p,q} is isomorphic to G_{p}^{p+q}/G_{p+1}^{p+q}.

Exercise 8.6. Consider two spectral sequences E and ′E that converge to G and G′ respectively. Let f: E → ′E be a morphism of spectral sequences and g: G → G′ be a map such that f_{∞}^{p,q}: E_{∞}^{p,q} → ′E_{∞}^{p,q} coincides with the map induced by g. Prove that if the map f_{r}^{p,q}: E_{r}^{p,q} → ′E_{r}^{p,q} for some r is an isomorphism, then g is an isomorphism too.

Now we describe an important method for constructing spectral sequences.

Definition 8.2. An exact couple is a pair of vector spaces (D, E) together with mappings i, j, k, such that the diagram

\begin{matrix}D & ⟶^{i} & D \\ _{k}↖ & ↙_{j} \\ E \end{matrix}

is exact in each vertex.

Set d = jk: E → E. Clearly, d^{2} = 0, so that we can define cohomology H(E, d) with respect to d. Given an exact couple, one defines the derived couple

\begin{matrix}D′ & ⟶^{i′} & D′ \\ _{k′}↖ & ↙_{j′} \\ E′ \end{matrix}

as follows: D′ = im i, E′ = H(E, d), i′ is the restriction of i to D′, j′(i(x)) for x ∈ D is the cohomology class of j(x) in H(E), the map k′ takes a cohomology class [y], y ∈ E, to the element k(y) ∈ D′.

Exercise 8.7. Check that mappings i′, j′, and k′ are well defined and that the derived couple is an exact couple.

Thus, starting from an exact couple C_{1} = (D, E, i, j, k) we obtain the sequence of exact couples C_{r} = (D_{r}, E_{r}, i_{r}, j_{r}, k_{r}) such that C_{r+1} is the derived couple for C_{r}.

A direct description of C_{r} in terms of C_{1} is as follows.

Proposition 8.4. The following isomorphisms hold for all r:

D_{r} = im i^{r−1},
E_{r} = k^{−1}(im i^{r−1})/j(ker i^{r−1}).

The map i_{r} is the restriction of i to D_{r}, j_{r}(i^{r−1}(x)) = [j(x)], and k_{r}([y]) = k(y), where [ · ] denotes equivalence class modulo j(ker i^{r−1}).

Proof. The proof is by induction on r and is left to the reader.

Now suppose that the exact couple C_{1} is bigraded, i.e., D = ⨁_{p,q} D^{p,q}, E = ⨁_{p,q} E^{p,q}, and the maps i, j, and k have bidegrees (−1, 1), (0, 0), (1, 0) respectively. In other words, one has:

i^{p,q}: D^{p,q} → D^{p−1,q+1},
j^{p,q}: D^{p,q} → E^{p,q},
k^{p,q}: E^{p,q} → D^{p+1,q}.

It is clear that the derived couples C_{r} are bigraded as well, and the mappings i_{r}, j_{r}, and k_{r} have bidegrees (−1, 1), (r − 1, 1 − r), (1, 0) respectively. Therefore the differential d_{r} is a differential in E_{r} and has bidegree (r, 1−r). Thus, (E_{r}^{p,q}, d_{r}^{p,q}) is a spectral sequence.

Now, suppose we are given a complex K^{•} with a decreasing filtration K_{p}^{•}. Each short exact sequence

0 ⟶ K_{p+1}^{•} ⟶ K_{p}^{•} ⟶ K_{p}^{•}/K_{p+1}^{•} ⟶ 0

induces the corresponding long exact sequence:

⋯ ⟶^{k} H^{p+q}(K_{p+1}^{•}) ⟶^{i} H^{p+q}(K_{p}^{•}) ⟶^{j} H^{p+q}(K_{p}^{•}/K_{p+1}^{•})

⟶^{k} H^{p+q+1}(K_{p+1}^{•}) ⟶^{i} · · · .

Hence, setting D_{1}^{p,q} = H^{p+q}(K_{p}^{•}) and E_{1}^{p,q} = H^{p+q}(K_{p}^{•}/K_{p+1}^{•}) we obtain a bigraded exact couple, with mappings having bidegrees as above. Thus we assign a spectral sequence to a complex with a filtration.

Let us compute the spaces E_{r}^{p,q} in an explicit form. Consider the upper term k^{−1}(im i^{r−1}) from the expression for E_{r}^{p,q} (see Proposition 8.4 on the preceding page). An element of E_{1}^{p,q} is a class [x] ∈ H^{p+q}(K_{p}^{•}/K_{p+1}^{•}), x ∈ K_{p}^{p+q}, dx ∈ K_{p+1}^{p+q}. The class [x]lies in k^{−1}(im i^{r−1}), if k([x]) ∈ H^{p+q+1}(K_{p+r}^{•}) ⊂ H^{p+q+1}(K_{p+1}^{•}). This is equivalent to dx = y + dz, with y ∈ K_{p+r}^{p+q}, z ∈ K_{p+1}^{p+q}. Thus, we see that x = (x−z)+z, with d(x−z) ∈ K_{p+r}^{p+q}. Denoting

Z_{r}^{p,q} = { w ∈ K_{p}^{p+q} | dw ∈ K_{p+r}^{p+q} },

we obtain k^{−1}(im i^{r−1}) = Z_{r}^{p,q} + K_{p+1}^{p+q}.

Further, consider the lower term j(ker i^{r−1}) from the expression for E_{r}^{p,q}. The kernel of the map i^{r−1} : H^{p+q}(K_{p}^{•}) → H^{p+q}(K_{p−r+1}^{•}) consists of cocycles x ∈ K_{p}^{p+q} such that x = dy for y ∈ K_{p−r+1}^{p+q−1}. So y ∈ Z_{r−1}^{p−r+1,q+r−2} and ker i^{r−1} = dZ_{r−1}^{p−r+1,q+r−2}. Then j(ker i^{r−1}) = dZ_{r−1}^{p−r+1,q+r−2} + K_{p+1}^{p+q}.

Thus, we get

E_{r}^{p,q} = \frac{Z_{r}^{p,q} + K_{p+1}^{p+q}}{dZ_{r−1}^{p−r+1,q+r−2} + K_{p+1}^{p+q}} = \frac{Z_{r}^{p,q}}{dZ_{r−1}^{p−r+1,q+r−2} + Z_{r−1}^{p+1,q−1}} .

Remark 8.3. The last equality follows from the well known Noether modular isomorphism

\frac{M + N}{M_{1} + N} = \frac{M}{M_{1} + (M ∩ N)}, M_{1} ⊂ M.

Theorem 8.5. If the filtration of the complex K^{•} is regular, then the spectral sequence of this complex converges to H^{•}(K^{•}) endowed with the filtration H_{p}^{k}(K^{•}) = im H^{k}(i_{p}), where i_{p}: K_{p}^{•} → K^{•} is the natural inclusion.

Proof. Note first, that if the filtration of the complex K^{•} is regular, then the spectral sequence of this complex is regular too. Further, the spaces C_{∞}^{p,q} and B_{∞}^{p,q} (see (8.6) on page 137) can easily be described by

C_{∞}^{p,q} = \frac{Z_{∞}^{p,q}}{Z_{∞}^{p+1,q−1}} , B_{∞}^{p,q} = \frac{(K_{p}^{p+q} ∩ d(K^{p+q−1})) + Z_{∞}^{p+1,q−1}}{Z_{∞}^{p+1,q−1}} ,

where Z_{∞}^{p,q} = { w ∈ K_{p}^{p+q} | dw = 0 }, whence

E_{∞}^{p,q} = \frac{Z_{∞}^{p,q}}{(K_{p}^{p+q} ∩ d(K^{p+q−1})) + Z_{∞}^{p+1,q−1}} .

Since H_{p}^{p+q}(K^{•}) = \frac{Z_{∞}^{p,q} + d(K^{p+q−1})}{d(K^{p+q−1})}, we have

\frac{H_{p}^{p+q}(K^{•})}{H_{p+1}^{p+q}(K^{•})} = \frac{Z_{∞}^{p,q} + d(K^{p+q−1})}{Z_{∞}^{p+1,q−1} + d(K^{p+q−1})}

= \frac{Z_{∞}^{p,q}}{Z_{∞}^{p+1,q−1} + (K_{p}^{p+q} ∩ d(K^{p+q−1}))} = E_{∞}^{p,q}.

This concludes the proof.

Definition 8.3. A bicomplex is a family of vector spaces K^{•,•} and linear mappings d′: K^{p,q} → K^{p+1,q}, d′′: K^{p,q} → K^{p,q+1}, such that (d′)^{2} = 0, (d′′)^{2} = 0, and d′d′′ + d′′d′ = 0.

Let K^{•} be the total (or diagonal) complex of a bicomplex K^{•,•}, i.e., by definition, K^{i} = ⨁_{i=p+q} K^{p,q} and d_{K} = d′ + d′′. There are two obvious filtration of K^{•}:

filtration I: ′K_{p}^{i} = ⨁_{\begin{matrix}j+q=i \\ j≥p \end{matrix}} K^{j,q},

filtration II: ′′K_{q}^{i} = ⨁_{\begin{matrix}p+j=i \\ j≥q \end{matrix}} K^{p,j}.

These two filtrations yield two spectral sequences, denoted respectively by ′E_{r}^{p,q} and ′′E_{r}^{p,q}.

It is easy to check that ′E_{1}^{p,q} = ′′H^{q}(K^{p,•}) and ′′E_{1}^{p,q} = ′H^{q}(K^{•,p}), where ′H (resp., ′′H) denotes the cohomology with respect to d′ (resp., d′′), with the differential d_{1} being induced respectively by d′ and d′′. Thus, we have:

Proposition 8.6. ′E_{2}^{p,q} = ′H^{p}(′′H^{q}(K^{•,•})) and ′′E_{2}^{p,q} = ′′H^{p}(′H^{q}(K^{•,•})).

Now assume that both filtrations are regular.

Exercise 8.8. Prove that

(1) if K^{p,q} = 0 for q < q_{0} (resp., p < p_{0}), then the first (resp., second) filtration is regular;

(2) if K^{p,q} = 0 for q < q_{0} and q > q_{1}, then both filtration are regular.

In this case both spectral sequences converge to the common limit H^{•}(K^{•}).

Remark 8.4. This fact does not mean that both spectral sequences have a common infinite term, because the two filtrations of H^{•}(K^{•}) are different.

Let us illustrate Proposition 8.6.

Example 8.7. Consider the commutative diagram

⋮  ⋮  ⋮
↑  ↑  ↑
0 ⟶ K^{2,0} ⟶^{d_{2}} K^{2,1} ⟶^{d_{2}} K^{2,2} ⟶ ⋯
↑^{d_{1}}  ↑^{d_{1}}  ↑^{d_{1}}
0 ⟶ K^{1,0} ⟶^{d_{2}} K^{1,1} ⟶^{d_{2}} K^{1,2} ⟶ ⋯
↑^{d_{1}}  ↑^{d_{1}}  ↑^{d_{1}}
0 ⟶ K^{0,0} ⟶^{d_{2}} K^{0,1} ⟶^{d_{2}} K^{0,2} ⟶ ⋯
↑  ↑  ↑
0  0  0

and suppose that the differential d_{1} is exact everywhere except for the terms K^{0,q} in the bottom row, and the differential d_{2} is exact everywhere except for the terms K^{p,0} in the left column. Thus, we have two complexes L_{1}^{•} and L_{2}^{•}, where L_{1}^{i} = H^{0}(K^{i,•}, d_{2}), L_{2}^{i} = H^{0}(K^{•,i}, d_{1}) and the differential of L_{1} (resp., L_{2}) is induced by d_{1} (resp., d_{2}). Consider the bicomplex K^{•,•} with (d′)^{p,q} = d_{1}^{p,q}, (d′′)^{p,q} = (−1)^{q}d_{2}^{p,q}. We easily get

′E_{2}^{p,q} = ′E_{3}^{p,q} = ⋯ = ′E_{∞}^{p,q} = \begin{cases}0 & if q ≠ 0, \\ H^{p}(L_{1}^{•}) & if q = 0, \end{cases}

′′E_{2}^{p,q} = ′′E_{3}^{p,q} = ⋯ = ′′E_{∞}^{p,q} = \begin{cases}0 & if p ≠ 0, \\ H^{q}(L_{2}^{•}) & if p = 0. \end{cases}

Since both spectral sequences converge to a common limit, we conclude that H^{i}(L_{1}^{•}) = H^{i}(L_{2}^{•}).

Let us describe this isomorphism in an explicit form. Consider a cohomology class from H^{i}(L_{1}^{•}). Choose an element k^{i,0} ∈ K^{i,0}, d_{1}(k^{i,0}) = 0, d_{2}(k^{i,0}) = 0, that represents this cohomology class. Since d_{1}(k^{i,0}) = 0, there exists an element x ∈ K^{i−1,0} such that d_{1}(x) = k^{i,0}. Set k^{i−1,1} = −d_{2}(x) ∈ K^{i−1,1}. We have d_{2}(k^{i−1,1}) = 0 and d_{1}(k^{i−1,1}) = −d_{1}(d_{2}(x)) = −d_{2}(d_{1}(x)) = −d_{2}(k^{i,0}) = 0. Further, the elements k^{i,0} and k^{i−1,1} are cohomologous in the total complex K^{•}: k^{i,0} − k^{i−1,1} = d_{1}x + d_{2}x = (d′ + d′′)(x). Continuing this process we obtain elements k^{i−j,j} ∈ K^{i−j,j}, d_{1}(k^{i−j,j}) = 0, d_{2}(k^{i−j,j}) = 0, that are cohomologous in the total complex K^{•}. Thus, the above isomorphism takes the cohomology class of k^{i,0} to that of k^{0,i}.

Exercise 8.9. Discuss an analog of Example 8.7 on the preceding page for the commutative diagram

⋮  ⋮  ⋮
↓  ↓  ↓
0 ⟵ K^{2,0} ⟵^{d_{2}} K^{2,1} ⟵^{d_{2}} K^{2,2} ⟵ ⋯
↓^{d_{1}} ↓^{d_{1}} ↓^{d_{1}}
0 ⟵ K^{1,0} ⟵^{d_{2}} K^{1,1} ⟵^{d_{2}} K^{1,2} ⟵ ⋯
↓^{d_{1}} ↓^{d_{1}} ↓^{d_{1}}
0 ⟵ K^{0,0} ⟵^{d_{2}} K^{0,1} ⟵^{d_{2}} K^{0,2} ⟵ ⋯
↓  ↓  ↓
0  0  0

References

[1] I. M. Anderson, Introduction to the variational bicomplex, Mathematical Aspects of Classical Field Theory (M. Gotay, J. E. Marsden, and V. E. Moncrief, eds.), Contemporary Mathematics, vol. 132, Amer. Math. Soc., Providence, RI, 1992, pp. 51–73.

[2] , The variational bicomplex, to appear.

[3] G. Barnich, Brackets in the jet-bundle approach to field theory, In Henneaux et al. [19], E-print hep-th/9709164.

[4] G. Barnich, F. Brandt, and M. Henneaux, Local BRST cohomology in Einstein-Yang-Mills theory, Nuclear Phys. B 445 (1995), 357–408, E-print hep-th/9505173.

[5] , Local BRST cohomology in the antifield formalism: I. General theorems, Comm. Math. Phys. 174 (1995), 57–92, E-print hep-th/9405109.

[6] , Local BRST cohomology in the antifield formalism: II. Application to Yang-Mills theory, Comm. Math. Phys. 174 (1995), 93–116, E-print hep-th/9405194.

[7] R. Bott and L. W. Tu, Differential forms in algebraic topology, Springer-Verlag, New York, 1982.

[8] N. Bourbaki, Algèbre. Chapitre X. Algèbre homologique, Masson, Paris, 1980.

[9] F. Brandt, Gauge covariant algebras and local BRST cohomology, In Henneaux et al. [19], E-print hep-th/9711171.

[10] R. L. Bryant, S.-S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Springer-Verlag, New York, 1991.

[11] R. L. Bryant and P. A. Griffiths, Characteristic cohomology of differential systems, I: General theory, J. Amer. Math. Soc. 8 (1995), 507–596, URL: http://www.math.duke.edu/˜ bryant.

[12] E. Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques, Hermann, Paris, 1946.

[13] A. Frölicher and A. Nijenhuis, Theory of vector valued differential forms. Part I: derivations in the graded ring of differential forms, Indag. Math. 18 (1956), 338– 359.

[14] M. Gerstenhaber and S. D. Schack, Algebraic cohomology and deformation theory, Deformation Theory of Algebras and Structures and Applications (M. Gerstenhaber and M. Hazewinkel, eds.), Kluwer, Dordrecht, 1988, pp. 11–264.

[15] D. M. Gessler, The “three-line” theorem for the Vinogradov C-spectral sequence of the Yang-Mills equations, Preprint SISSA 71/95/FM, 1995, URL: http://ecfor.rssi.ru/˜ diffiety/.

[16] , On the Vinogradov C-spectral sequence for determined systems of differential equations, Differential Geom. Appl. 7 (1997), 303–324, URL: http://ecfor.rssi.ru/˜ diffiety/.

[17] H. Goldschmidt, Existence theorems for analytic linear partial differential equations, Ann. of Math. (2) 86 (1967), 246–270.

[18] M. Henneaux, Consistent interactions between gauge fields: The cohomological approach, In Henneaux et al. [19], E-print hep-th/9712226.

[19] M. Henneaux, I. S. Krasil′shchik, and A. M. Vinogradov (eds.), Secondary calculus and cohomological physics, Contemporary Mathematics, vol. 219, Amer. Math. Soc., Providence, RI, 1998.

[20] P. J. Hilton and U. Stammbach, A course in homological algebra, Springer-Verlag, Berlin, 1971.

[21] N. G. Khor′kova, Conservation laws and nonlocal symmetries, Math. Notes 44 (1989), 562–568.

[22] K. Kiso, Pseudopotentials and symmetries of evolution equations, Hokkaido Math. J. 18 (1989), 125–136.

[23] Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 153–165.

[24] I. S. Krasil′shchik, Some new cohomological invariants for nonlinear differential equations, Differential Geom. Appl. 2 (1992), 307–350.

[25] , Hamiltonian formalism and supersymmetry for nonlinear differential equations, Preprint ESI 257, 1995, URL: http://www.esi.ac.at.

[26] , Calculus over commutative algebras: A concise user guide, Acta Appl. Math. 49 (1997), 235–248, URL: http://ecfor.rssi.ru/˜ diffiety/.

[27] , Algebras with flat connections and symmetries of differential equations, Lie Groups and Lie Algebras: Their Representations, Generalizations and Applications (B. P. Komrakov, I. S. Krasil′shchik, G. L. Litvinov, and A. B. Sossinsky, eds.), Kluwer, Dordrecht, 1998, pp. 425–434.

[28] , Cohomology background in the geometry of PDE, In Henneaux et al. [19], URL: http://ecfor.rssi.ru/˜ diffiety/.

[29] I. S. Krasil′shchik and P. H. M. Kersten, Deformations and recursion operators for evolution equations, Geometry in Partial Differential Equations (A. Prastaro and Th. M. Rassias, eds.), World Scientific, Singapore, 1994, pp. 114–154.

[30] , Graded differential equations and their deformations: A computational theory for recursion operators, Acta Appl. Math. 41 (1994), 167–191.

[31] , Graded Frölicher-Nijenhuis brackets and the theory of recursion operators for super differential equations, The Interplay between Differential Geometry and Differential Equations (V. V. Lychagin, ed.), Amer. Math. Soc. Transl. (2), Amer. Math. Soc., Providence, RI, 1995, pp. 143–164.

[32] I. S. Krasil′shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach, New York, 1986.

[33] I. S. Krasil′shchik and A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1989), 161–209.

[34] I. S. Krasil′shchik and A. M. Vinogradov (eds.), Symmetries and conservation laws for differential equations of mathematical physics, Monograph, Faktorial Publ., Moscow, 1997 (Russian; English transl. will be published by the Amer. Math. Soc.).

[35] V. V. Lychagin, Singularities of multivalued solutions of nonlinear differential equations, and nonlinear phenomena, Acta Appl. Math. 3 (1985), 135–173.

[36] V. V. Lychagin and V. N. Rubtsov, Local classification of Monge-Ampère equations, Soviet Math. Dokl. 28 (1983), 328–332.

[37] S. MacLane, Homology, Springer-Verlag, Berlin, 1963.

[38] B. Malgrange, Ideals of differentiable functions, Oxford University Press, Oxford, 1966.

[39] M. Marvan, On the C-spectral sequence with “general” coefficients, Differential Geometry and Its Applications, Proc. Conf. Brno, 1989, World Scientific, Singapore, 1990, pp. 361–371.

[40] , On zero-curvature representations of partial differential equations, Differential Geometry and Its Applications, Proc. Conf. Opava, 1992, Open Education and Sciences, Opava, 1993, pp. 103–122, URL: http://www.emis.de/proceedings/.

[41] J. McCleary, A user’s guide to spectral sequences, Publish or Perish, Inc., Wilmington, Delaware, 1985.

[42] A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants for certain tensor fields, I, Indag. Math. 17 (1955), 390–403.

[43] P. J. Olver, Applications of Lie groups to differential equations, 2nd ed., Springer-Verlag, New York, 1993.

[44] D. Hernández Ruipérez and J. Muñoz Masqué, Global variational calculus on graded manifolds, I: graded jet bundles, structure 1-form and graded infinitesimal contact transformations, J. Math. Pures Appl. 63 (1984), 283–309.

[45] , Global variational calculus on graded manifolds, II, J. Math. Pures Appl. 64 (1985), 87–104.

[46] D. C. Spencer, Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179–239.

[47] T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), 311–363.

[48] , Formal geometry of systems of differential equations, Sugaku Expositions 3 (1990), 25–73.

[49] , Homological method of computing invariants of systems of differential equations, Differential Geom. Appl. 1 (1991), 3–34.

[50] W. M. Tulczyjew, The Lagrange complex, Bull. Soc. Math. France 105 (1977), 419– 431.

[51] , The Euler-Lagrange resolution, Differential Geometrical Methods in Mathematical Physics (P. L. García, A. Pérez-Rendón, and J.-M. Souriau, eds.), Lecture Notes in Math., no. 836, Springer-Verlag, Berlin, 1980, pp. 22–48.

[52] , Cohomology of the Lagrange complex, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 14 (1987), 217–227.

[53] A. M. Verbovetsky, Lagrangian formalism over graded algebras, J. Geom. Phys. 18 (1996), 195–214, E-print hep-th/9407037.

[54] , Differential operators over quantum spaces, Acta Appl. Math. 49 (1997), 339–361, URL: http://ecfor.rssi.ru/˜ diffiety/.

[55] , On quantized algebra of Wess-Zumino differential operators at roots of unity, Acta Appl. Math. 49 (1997), 363–370, E-print q-alg/9505005.

[56] , Notes on the horizontal cohomology, In Henneaux et al. [19], E-print math.DG/9803115.

[57] A. M. Verbovetsky, A. M. Vinogradov, and D. M. Gessler, Scalar differential invariants and characteristic classes of homogeneous geometric structures, Math. Notes 51 (1992), 543–549.

[58] A. M. Vinogradov, The logic algebra for the theory of linear differential operators, Soviet Math. Dokl. 13 (1972), 1058–1062.

[59] , On algebro-geometric foundations of Lagrangian field theory, Soviet Math. Dokl. 18 (1977), 1200–1204.

[60] , A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints, Soviet Math. Dokl. 19 (1978), 144–148.

[61] , Geometry of nonlinear differential equations, J. Soviet Math. 17 (1981), 1624–1649.

[62] , Local symmetries and conservation laws, Acta Appl. Math. 2 (1984), 21–78.

[63] , The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory. II. The nonlinear theory, J. Math. Anal. Appl. 100 (1984), 1–129.

[64] A. M. Vinogradov and I. S. Krasil′shchik, A method of computing higher symmetries of nonlinear evolution equations and nonlocal symmetries, Soviet Math. Dokl. 22 (1980), 235–239.