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Preprint DIPS–3/98 January 26, 1998

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Notes on the horizontal cohomology

by

Alexander VERBOVETSKY

math.DG/9803115

Notes on the horizontal cohomology

Alexander VERBOVETSKY

Abstract. This paper is devoted to the horizontal (“characteristic”) cohomology of systems of differential equations. Recent results on computing the horizontal cohomology via the compatibility complex are generalized. New results on the Vinogradov C-spectral sequence and Krasil′shchik’s C-cohomology are obtained. As an application of general theory, the examples of an evolution equation and a p-form gauge theory are explicitly worked out.

Introduction

Consider a system of differential equations

F_{s}(x_{i}, u^{j}, …, u_{σ}^{j}, …) = 0,  (∗)

with independent variables x_{i}, unknown functions u^{j}, and u_{σ}^{j} = ∂^{r}u^{j}/∂x_{i_{1}} ⋯ ∂x_{i_{r}} being their derivatives, σ = i_{1} … i_{r}. Let F be the algebra of functions in the variables x_{i}, u^{j}, and u_{σ}^{j}. Two functions are said to be equivalent if they coincide when equations (∗) hold. Denote the quotient algebra by F_{F}. The horizontal de Rham complex for system (∗) is the exterior algebra generated by elements of F_{F} and differentials dx_{i}, with the differential being the total exterior differential d̄(f dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{k}}) = ∑_{i}(df/dx_{i}) dx_{i} ∧ dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{k}}. The cohomology of this complex, called the horizontal (“characteristic”) cohomology, has been studied and used extensively in the literature (see the bibliography). This cohomology plays a central role in the Lagrangian formalism (because the horizontal cohomology group in degree n is the space of actions of variational problems constrained by equations (∗)). The horizontal cohomology group in degree n−1 consists of conservation laws for the system under consideration. This fact is basic to finding conservation laws via the Vinogradov C-spectral sequence. The horizontal cohomology in degrees less than n − 1 has attracted recently a great deal of interest in the context of gauge theories (for equations without gauge symmetries this cohomology is trivial by the famous “two-line theorem” of Vinogradov [28, 30]), where it appears as a means of calculating the BRST cohomology [3].

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 module of coefficients is supplied with a filtration such that the differential of the associated graded complex is linear over the functions. Hence, the cohomology can be computed algebraically. For p = 1, this has been done in [23] and the main result is: The term E_{1}^{1,•} coincides with the homology of the complex dual to the compatibility complex for the linearization operator of the system under consideration. In this paper, this result is generalized to p > 1. We compute also Krasil′shchik’s C-cohomology (see [9, 10, 11, 12, 13, 14]). In particular, the “k-line theorem” is proved for both cases. It should be also stressed that our techniques are capable not only of obtaining vanishing results, but facilitate the computation of nonzero part as well.

Talk presented at the conference “Secondary Calculus and Cohomological Physics” in Moscow, August 1997. It will be published in the Proceedings of this conference (M. Henneaux, I. S.

Krasil′shchik, and A. M. Vinogradov, eds.) in Contemporary Mathematics, Amer. Math. Soc.

This work was supported in part by RFBR grant 97-01-00462 and INTAS grant 96-0793.

The author’s thinking about computing the horizontal cohomology was especially stimulated by works of I. S. Krasil′shchik [9] and M. Marvan [19] where the horizontal cohomology with coefficients in the Cartan forms and zero-curvature representations was calculated. By trying to understand their calculations, the author arrived at the construction of the present paper.

The ability to calculate the horizontal cohomology is not unique to the method described above. There is also an approach based on the Koszul - Tate resolution. The discussion of this approach does not enter into the scope of the present paper. It is desired here merely to point out that both methods take as a starting point the compatibility complex and furnish equivalent results although the precise relationships between them have yet to be uncovered.

The paper is organized as follows. We begin with a short introduction to the C-differential calculus on differential equations. In the opening subsection of Section 2, we discuss the notion of a C-module. A C-module is, by definition, a module equipped with an action of C-differential operators. Thus, C-modules serve as modules of coefficients for horizontal de Rham complexes. Such complexes are dealt with in Section 2.2. In Section 2.3, we relate the horizontal cohomology to the cohomology of the compatibility complex. The results of this section play a key role in the explicit calculation of the horizontal cohomology. Finally, in Section 3 we turn to the examples of computing the Vinogradov C-spectral sequence and Krasil′shchik’s C-cohomology.

The author would like to thank Dmitri Gessler, Joseph S. Krasil′shchik, and Alexander M. Vinogradov for many useful discussions and collaborations. It is a pleasure to thank the organizers of the International Conference on Secondary Calculus and Cohomological Physics in Moscow for the invitation, providing a stimulating environment, and financial help. He is also grateful to the Scuola Internazionale Superiore di Studi Avanzati in Trieste for its kind hospitality during the author’s visit in 1993–95 when a part of this research was done.

1. C-differential calculus

1.1. Jet spaces. In this section a brief discussion of the C-differential calculus on differential equations is given, further details being available in [17, 15, 29, 20].

Let π: E → M be an n+m-dimensional vector bundle over an n-dimensional base manifold M, and let π_{∞}: J^{∞}(π) → M be the infinite jet bundle of local sections of π. Denote by j_{∞}(s)(x) the infinite jet of a local section s of the bundle π at point x ∈ M. Thus each section s: M → E gives rise to the section j_{∞}(s): M → J^{∞}(π).

A coordinate system (x_{i}, u^{j}), i = 1, …, n, j = 1, …, m, of π: E → M induces coordinates (x_{i}, u^{j}, u_{i}^{j}, …, u_{σ}^{j}, …) on J^{∞}(π) such that

u_{σ}^{j}(j_{∞}(s)(x)) = \frac{∂^{|σ|}s^{j}}{∂x_{i_{1}}∂x_{i_{2}} ⋯ ∂x_{i_{r}}} ,

where σ = i_{1}i_{2} … i_{r} is a multi-index, |σ| = r.

Denote the algebra of functions on J^{∞}(π) by F = F(π). Functions on J^{∞}(π) are smooth functions of a finite number of coordinates. Let α: W → J^{∞}(π) be a vector bundle and Γ(α) be the set of local sections of α. Naturally, Γ(α) is a module over the algebra F. In the sequel, by a module we always mean a module of this form. Recall that homomorphisms of modules are in one-to-one correspondence with morphisms of bundles, so we can and will not distinguish between bundles and modules.

On an infinite jet space, there exists the C-differential calculus. In coordinate language, this is the total derivatives calculus. Let us describe C-differential operators, corresponding jets, and forms in more details.

Consider two F-modules P and P_{1}. A differential operator Δ: P → P_{1} is called C-differential, if it can be restricted to the manifolds of the form j_{∞}(s)(M), where s is a local section of π. In other words, Δ is a C-differential operator, if the equality j_{∞}(s)^{∗}(φ) = 0, φ ∈ P, implies j_{∞}(s)^{∗}(Δ(φ)) = 0.

In local coordinates, C-differential operators have the form

(\begin{matrix}∑_{σ} a_{11}^{σ}D_{σ} & & … & & & ∑_{σ} a_{1m_{1}}^{σ}D_{σ} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{σ} a_{m_{2}1}^{σ}D_{σ} & & … & & & ∑_{σ} a_{m_{2}m_{1}}^{σ}D_{σ} \end{matrix}) ,

where a_{ij}^{σ} ∈ F, D_{σ} = D_{i_{1}}∘⋯∘D_{i_{r}} for σ = i_{1} … i_{r}, and D_{i} = ∂/∂x_{i}+∑_{j,σ} u_{σi}^{j}∂/∂u_{σ}^{j} is the i-th total derivative operator.

We shall denote the F-module of C-differential operators from P to P_{1} by CDiff(P, P_{1}). In this module, there exists a filtration by the modules CDiff_{k}(P, P_{1}) that consist of C-differential operators of order ≤ k.

Next, define the module of horizontal jets. Let P be an F-module. Say that two elements of P are horizontally equivalent up to order k ≤ ∞ at point θ ∈ J^{∞}(π), 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 ∈ J^{∞}(π)} J̄_{θ}^{k}(P) constitutes the horizontal jet bundle J̄^{k}(P) → J^{∞}(π). 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}. For any C-differential operator Δ: P → P_{1} of order ≤ k, there is a unique F-homomorphism φ_{Δ} : J̄^{k}(P) → P_{1} such that Δ = φ_{Δ} ∘ j̄_{k}. The composition

j̄_{l} ∘ Δ ∈ CDiff_{k+l}(P, J̄^{l}(P_{1}))

is called the l-th prolongation of Δ and the corresponding homomorphism from J̄^{k+l}(P) to J̄^{l}(P_{1}) is denoted by φ_{Δ}^{l}. In particular, Δ gives rise to the mapping φ_{Δ}^{∞}: J̄^{∞}(P) → J̄^{∞}(P_{1}).

Let D(π) be the F-module of vector fields on J^{∞}(π). Define the module of Cartan vector fields as the intersection CD(π) = D(π) ∩ CDiff_{1}(F, F). In coordinates, a Cartan vector field is ∑_{i} f_{i}D_{i}, f_{i} ∈ F. A vector field X ∈ D(π) is called vertical if X(j_{∞}(s)^{∗}(h)) = 0 for all functions h ∈ C^{∞}(M). Locally a vertical vector field has the form ∑_{j,σ} f_{j,σ}∂/∂u_{σ}^{j}. Denote the module of vertical vector fields by D^{v}(π). Clearly, D(π) = CD(π) ⊕ D^{v}(π).

Further, consider the module Λ^{k}(π) of differential k-forms on J^{∞}(π). A form ω ∈ Λ^{k}(π) is called a Cartan form, if ω satisfies j_{∞}(s)^{∗}(ω) = 0 for every local section s of π. Equivalently, Cartan forms are forms that vanish on the Cartan vector fields. The set of all Cartan forms defines an ideal

CΛ^{∗}(π) = ⨁_{k ≥ 0} CΛ^{k}(π)

in the ring

Λ^{∗}(π) = ⨁_{k ≥ 0} Λ^{k}(π)

of all forms on J^{∞}(π). In coordinates, this ideal is generated by the Cartan 1-forms

ω_{σ}^{j} = du_{σ}^{j} − ∑_{i} u_{σi}^{j} dx_{i}

for all j and σ. The quotient algebra

Λ̄^{∗}(π) = Λ^{∗}(π)/CΛ^{∗}(π)

is called the algebra of horizontal forms on J^{∞}(π). In coordinates, a horizontal k-form is a sum of terms of the form

f dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{k}}, f ∈ F(π).

The exterior derivative

d: Λ^{k}(π) → Λ^{k+1}(π)

gives rise to the horizontal differential

d̄: Λ̄^{k}(π) → Λ̄^{k+1}(π),

since the ideal of Cartan forms CΛ^{∗}(M) is stable with respect to d: d(CΛ^{∗}(π)) ⊂ CΛ^{∗}(π). Thus we get the horizontal de Rham complex

0 ⟶ F(π) ⟶^{d̄} Λ̄^{1}(π) ⟶^{d̄} Λ̄^{2}(π) ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n}(π) ⟶ 0.

More general, consider the filtration in the de Rham complex on the jet space J^{∞}(π)

⋯ ⊂ C^{k+1}Λ^{∗} ⊂ C^{k}Λ^{∗} ⊂ ⋯ ⊂ C^{2}Λ^{∗} ⊂ CΛ^{∗} ⊂ Λ^{∗},

where C^{k}Λ^{∗} is the k-th exterior power of the Cartan ideal CΛ^{∗}. Evidently, d(C^{k}Λ^{∗}) ⊂ C^{k}Λ^{∗}, so that we obtain a spectral sequence converging to the de Rham cohomology of J^{∞}(π). The zero term of the spectral sequence is E_{0}^{p,q} = C^{p}Λ^{p+q} = CΛ^{1} ∧ ⋯ ∧ CΛ^{1} ⊗ Λ̄^{q}. This spectral sequence is called the Vinogradov C-spectral sequence [27, 28]. Further details on this important spectral sequence can be found in [30, 17, 31, 21, 22, 23, 5, 1, 2, 26, 18] and below in this paper as well.

Given a C-differential operator Δ ∈ CDiff_{k}(P, P_{1}), we define the (l-th) symbol σ(Δ) of Δ by the following commutative diagram

0 ⟶ S^{k+l}(Λ̄^{1}) ⊗ P ⟶ J̄^{k+l}(P) ⟶^{ν_{k+l,k+l−1}} J̄^{k+l−1}(P) ⟶ 0
↓^{σ(Δ)}  ↓^{φ_{Δ}^{k+l}}  ↓^{φ_{Δ}^{k+l−1}}
0 ⟶ S^{k}(Λ̄^{1}) ⊗ P_{1} ⟶ J̄^{k}(P_{1}) ⟶^{ν_{k,k−1}}  J̄^{k−1}(P_{1}) ⟶ 0.

The rows of the diagram are exact, with inclusions S^{k}(Λ̄^{1}) ⊗ P → J̄^{k}(P) given by df_{1} ⋯ df_{k} ⊗ p ↦ [… [j̄_{k}, f_{1}], …, f_{k}](p), where p ∈ P, [· , f_{i}] is the commutator with the operator of multiplication by f_{i} ∈ F.

Let us consider the pullback of π along the projection J^{∞}(π) → M and denote the module of sections of this vector bundle by κ. It is readily seen that for any point θ = j_{∞}(s)(x) ∈ J^{∞}(π) one has

T_{θ}(J^{∞}(π)) = J_{x}^{∞}(π) = J̄_{θ}^{∞}(κ).  (1.1)

This yields the canonical isomorphism

D^{v}(π) = J̄^{∞}(κ).  (1.2)

The dual isomorphism reads

CΛ^{1}(π) = CDiff(κ, F).  (1.3)

In coordinates, the form ω_{σ}^{j} under this isomorphism is the operator (…, D_{σ}, …), with D_{σ} on j-th place.

It is clear that the Cartan k-forms can be identified with multilinear skewsymmetric C-differential operators in k arguments.

1.2. Differential equations. Pick up a system of k-th order partial differential equations

F_{s}(x_{i}, u^{j}, …, u_{σ}^{j}, …) = 0, s = 1, …, l.  (1.4)

We shall consider F = (F_{1}, …, F_{l}) as an element of an F-module P. The system F = 0 defines a subbundle E → M of the jet bundle J^{k}(π) → M. A solution to the system of differential equations is a section s: M → E such that j_{k}(s)(M) ⊂ E. One can define the infinite prolongation E^{∞} of E by equations F_{s} = 0, …, D_{σ}F_{s} = 0, … This system is equivalent to (1.4) in the sense that both have the same set of solutions. For brevity, E and E^{∞} will be referred to as “equation”.

We define the function algebra F(E) on E^{∞} to be the restriction of the algebra F(π) on J^{∞}(π). It is straightforward to show that one can pull all ingredients of the C-differential calculus on J^{∞}(π), discussed in the previous subsection, back to E^{∞}. Thus, we have only one thing to do here: to generalize formulae (1.2) and (1.3).

Pick a point θ = j_{∞}(s)(x) ∈ E^{∞}. In view of (1.1), the tangent space T_{θ}(E^{∞}) is isomorphic to a subspace R_{θ} ⊂ J̄^{∞}(κ). In coordinates, the subspace T_{θ}(E^{∞}) ⊂ T_{θ}(J^{∞}(π)) is given by the equations ∑_{j,σ}(∂D_{τ} (F_{s})/∂u_{σ}^{j}) du_{σ}^{j} = 0, τ being a multiindex. Therefore R_{θ} is defined by

∑_{j,σ} \frac{∂D_{τ} (F_{s})}{∂u_{σ}^{j}}w_{σ}^{j} = 0,  (1.5)

where w_{σ}^{j} = π_{∞}^{∗}(u_{σ}^{j}) are coordinates on J̄^{∞}(κ). Clearly, system (1.5) can be rewritten in the form

∑_{j,σ} D_{τ} (\frac{∂F_{s}}{∂u_{σ}^{j}} w_{σ}^{j}) = 0

or D_{τ} (ℓ_{F} (w^{j})) = 0, where

ℓ_{F} = (\begin{matrix}∑_{σ} \frac{∂F_{1}}{∂u_{σ}^{1}} D_{σ} & & & … & & ∑_{σ} \frac{∂F_{1}}{∂u_{σ}^{m}} D_{σ} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ∑_{σ} \frac{∂F_{l}}{∂u_{σ}^{1}} D_{σ} & & & … & & ∑_{σ} \frac{∂F_{l}}{∂u_{σ}^{m}} D_{σ} \end{matrix})

is the operator of universal linearization for F.

Now our discussion can be summarized as follows.

Proposition 1.1. For any differential equation E,

1. The module D^{v}(E) is isomorphic to the kernel of the homomorphism φ_{ℓ_{F}}^{∞} : J̄^{∞}(κ) → J̄^{∞}(P).

2. The module CΛ^{1}(E) is isomorphic to CDiff(κ, F) modulo the submodule consisting of the operators of the form ∇ ∘ ℓ_{F}, ∇ ∈ CDiff(P, F).

We use the notation D^{v}(E), C^{k}Λ^{s}(E), and so on, for the corresponding modules on E^{∞}.

Remark 1.2. The constructions and results covered in this paper are valid not only for equations E^{∞}, but for arbitrary diffieties [31, 17] as well. Recall, that a diffiety is an infinite-dimensional manifold furnished with an involutive finite-dimensional distribution that locally is of the form E^{∞} endowed with the distribution of Cartan fields.

2. Horizontal cohomology

2.1. C-modules on differential equations. Fix an equation E^{∞}. Let F = F(E) be the algebra of functions on E^{∞}. 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-differential 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}.

In other words, 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) of Cartan vector fields, X ↦ ∇_{X}, that is F-linear:

∇_{fX+gY} = f∇_{X} + g∇_{Y} , f, g ∈ F, X, Y ∈ CD,

satisfies the Leibnitz 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]}.

The coordinate description of a flat horizontal connection looks as

∇_{D_{i}}(s_{j}) = ∑_{k} Γ_{ij}^{k}s_{k}, Γ_{ij}^{k} ∈ F,

where s_{j} are basis elements of Q.

Remark 2.1. Let Q be the module of sections of a vector bundle τ: W → E^{∞}, Q = Γ(τ). A flat horizontal connection on Q defines a completely integrable ndimensional linear distribution on W that is projected onto the Cartan distribution on E^{∞}. Thus, geometrically, a C-module is the module of sections of a linear covering (see [17, 16]).

In coordinates the covering has the form

D̃_{i} = D_{i} + ∑_{j,k} Γ_{ij}^{k}w^{j} \frac{∂}{∂w_{k}} ,

where w^{i} are fiber coordinate on W.

Here are basic examples of C-modules.

Example 2.2. The simplest example of a C-module is Q = F with the usual action of horizontal operators.

Example 2.3. The module of vertical vector fields Q = D^{v} = D^{v}(E) with the connection

∇_{X}(Y) = [Y, X], X ∈ CD, Y ∈ D^{v}

is a C-module.

Example 2.4. Next example is the modules of Cartan forms Q = C^{k}Λ^{k} = C^{k}Λ^{k}(E). A vector field X ∈ CD acts on C^{k}Λ^{k} as the Lie derivative L_{X}. It is easily seen that in coordinates we have

(D_{i})_{C^{k}Λ^{k}}(ω_{σ}^{j}) = ω_{σi}^{j}

Example 2.5. The infinite jet module Q = J̄^{∞}(P) of an F-module P is a C-mod-ule via

∇_{X}(fj̄_{∞}(p)) = X(f)j̄_{∞}(p),

where X ∈ CD, f ∈ F, p ∈ P.

Example 2.6. Let us dualize the previous example. It is clear that for any Fmodule P the module Q = CDiff(P, F) is a C-module. The action of horizontal operators is the composition.

Example 2.7. More generally, let Δ: P → P_{1} be a C-differential operator and φ_{Δ}^{∞}: J̄^{∞}(P) → J̄^{∞}(P_{1}) be the corresponding prolongation of Δ. Obviously, φ_{Δ}^{∞} is a morphism of C-modules, i.e., a homomorphism over the ring CDiff(F, F), so that ker φ_{Δ}^{∞} and coker φ_{Δ}^{∞} are C-modules.

Dually, the operator Δ gives rise to the morphism of C-modules CDiff(P_{1}, F) → CDiff(P, F), ∇ ↦ ∇ ∘ Δ. Thus the kernel and cokernel of this map are C-modules.

Example 2.8. 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}, f ∈ Hom_{F}(Q_{1}, Q_{2}).

For instance, we have C-module structures on Q = J̄^{∞}(P) ⊗_{F} C^{k}Λ^{k} and Q = CDiff(P, C^{k}Λ^{k}) for any F-module P.

Example 2.9. 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^{∞}. Such C-modules are called zero-curvature representations over E^{∞} (cf. [19]). 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}) , 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.

In coordinates, if the form ω is given by ω = ∑_{i} A_{i} dx_{i}, A_{i} ∈ 𝔤, then for any C-differential operator Δ the coordinate description of the operator Δ_{Q} can be obtained by replacing all occurrences of D_{i} with D_{i} + ad A_{i}.

Remark 2.10. In parallel with left C-modules one can consider right C-modules, 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.

Take a C-module Q. By definition, for a scalar C-differential operator Δ: F → F there exists the operator Δ_{Q}: Q → Q. In fact one has more:

Proposition 2.11. 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-differential 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.

Lemma 2.12 ([30]). 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 lemma, the family of operators

Δ_{Q}[p ⊗ q, s^{∗} ⊗ q^{∗}](f) = q^{∗}(Δ(p, s^{∗})_{Q}(fq))

uniquely determines the operator Δ_{Q}.

2.2. The horizontal de Rham complex. Consider a complex of C-differential 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 2.1, 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 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 chain 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.

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).

Let us discuss some examples of the horizontal de Rham complexes.

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−2}(P) ⟶ 0
↓δ̄  ↓S̄  ↓S̄
⋮  ⋮  ⋮

Diagram 2.1

Example 2.13. The horizontal de Rham complex with coefficients in J̄^{∞}(P)

0 ⟶ J̄^{∞}(P) ⟶^{d̄} Λ̄^{1} ⊗ J̄^{∞}(P) ⟶^{d̄} Λ̄^{2} ⊗ J̄^{∞}(P) ⟶^{d̄} ⋯ ⟶^{d̄} Λ̄^{n} ⊗ J̄^{∞}(P) ⟶ 0

turns out to be the project limit of the horizontal Spencer complexes

0 ⟶ J̄^{k}(P) ⟶^{S̄} Λ̄^{1} ⊗ J̄^{k−1}(P) ⟶^{S̄} Λ̄^{2} ⊗ J̄^{k−1}(P) ⟶^{S̄} ⋯ ⟶^{S̄} Λ̄^{n} ⊗ J̄^{k−n}(P) ⟶ 0, (2.1)

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.

The proof is standard but we sketch it now because it involves issues which will be needed further. Consider the short exact sequence of the complexes in Diagram 2.1, with S̄^{k} = S^{k}(Λ̄^{1}) being the symmetric power of Λ̄^{1}. The left column, called the horizontal Spencer δ-complex, is a complex of homomorphisms. The operator δ̄: Λ̄^{s} ⊗ S̄^{r} ⊗ P → Λ̄^{s+1} ⊗ S̄^{r−1} ⊗ P is defined by δ̄(ω ⊗ u ⊗ p) = (−1)^{s}ω ∧ i(u) ⊗ p, where i: S̄^{r} → Λ̄^{1} ⊗ S̄^{r−1} is the natural inclusion. Dropping the multiplier P and considering the sequence at a point of E^{∞}, we get the de Rham complex with polynomial coefficients. This proves that the Spencer δ-complex is exact. Hence the second and third columns have the same cohomology, so that the cohomology of (2.1) can be computed at k = 0 and the desired statement is proved.

Remark 2.14. Here we encounter the phenomenon mentioned in the Introduction: the existence of a filtration in the C-module of coefficients, such that the associated graded complex is a complex of F-homomorphisms, makes it possible to calculate the horizontal cohomology locally.

Now, let us multiply the previous diagram by a C-module Q (possibly of infinite rank). Arguing as above, we see 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 2.15. The dualization of the previous example is the following. 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 operator as follows (cf. [23, 24]). Any horizontal differential operator

Δ: P → P_{1}

gives rise to the morphism of complexes

0  0
↓  ↓
CDiff(P_{1}, F) ⟶ CDiff(P, F)
↓  ↓
CDiff(P_{1}, Λ̄^{1}) ⟶ CDiff(P, Λ̄^{1})
↓  ↓
CDiff(P_{1}, Λ̄^{2}) ⟶ CDiff(P, Λ̄^{2})
↓  ↓
⋮  ⋮
↓  ↓
CDiff(P_{1}, Λ̄^{n}) ⟶ CDiff(P, Λ̄^{n})
↓  ↓
0  0

and hence the map of the cohomology groups Δ^{∗}: P̂_{1} → P̂, which is the adjoint operator. The reader will have no difficulty in showing that

1. (Δ_{1} ∘ Δ_{2})^{∗} = Δ_{2}^{∗} ∘ Δ_{1}^{∗};

2. if Δ ∈ CDiff_{k}(P, P_{1}) then Δ^{∗} ∈ CDiff_{k}(P̂_{1}, P̂). In coordinates, we have (∑_{σ} f_{σ}D_{σ})^{∗} = (−1)^{|σ|} ∑_{σ} D_{σ} ∘ f_{σ} for a scalar operator and ∥Δ_{ij}∥^{∗} = ∥Δ_{ji}^{∗}∥ for a matrix one.

Example 2.16. The choice Q = Q_{0} ⊗ C^{p}Λ^{p}, with Q_{0} a zero-curvature representation, leads to the p-th gauge complex (cf. [19]).

Example 2.17. The zero term E_{0}^{p,q} of the Vinogradov C-spectral sequence consists of the complexes

0 ⟶ E_{0}^{p,0} ⟶ E_{0}^{p,1} ⟶ ⋯ ⟶ E_{0}^{p,n} ⟶ 0, p = 0, 1, 2, …,

which are the horizontal de Rham complexes with coefficients in C^{p}Λ^{p}.

Example 2.18. Take the C-module

Q = ⨁_{p} D^{v}(C^{p}Λ^{p}) = ⨁_{p} Hom_{F}(CΛ^{1}, C^{p}Λ^{p}).

The horizontal de Rham complex with coefficient in Q can be written as

0 ⟶ D^{v} ⟶ D^{v}(Λ^{1}) ⟶ D^{v}(Λ^{2}) ⟶ ⋯

It is simply a calculation to verify that this is the complex introduced by Krasil′- shchik [9]. The cohomology of this complex is called the C-cohomology. It contains important invariants of differential equations, in particular, symmetries and deformations of the Cartan structure (= recursion operators) (see [9, 10, 11, 12, 13, 14]).

The calculation of the cohomology of the complexes from the last two examples is our main concern in this paper.

2.3. Compatibility complex. Recall that 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.

Consider a C-differential operator Δ: P_{0} → P_{1} and the corresponding C-module R_{Δ} = ker φ_{Δ}^{∞} (cf. example 2.7). Suppose that there exists a formally exact complex

P_{0} ⟶^{Δ} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3} ⟶^{Δ_{3}} · · · .  (2.2)

Then the cohomology of this complex coincides with the horizontal cohomology with coefficients in R_{Δ}:

Theorem 2.19.

H̄^{i}(R_{Δ}) = H^{i}(P_{•}).

Proof. Consider the following 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

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 completes the proof.

Let us multiply the previous diagram by a C-module Q. This yields

H̄^{i}(R_{Δ} ˆ⊗ Q) = H^{i}(P_{•} ⊗ Q),  (2.3)

where R_{Δ} ˆ⊗ Q = proj lim R_{Δ}^{l} ⊗ Q, with R_{Δ}^{l} = ker φ_{Δ}^{k+l}, ord Δ ≤ k.

We can dualize our discussion. Namely, consider the 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),  (2.4)

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 (2.2).

Now, suppose we are given a C-differential operator Δ ∈ CDiff_{k}(P_{0}, P_{1}). How do we find a formally exact complex of the form (2.2)? To this end consider for each positive integer k_{1} the mapping φ_{Δ}^{k+k_{1}} : J̄^{k+k_{1}}(P_{0}) → J̄^{k_{1}}(P_{1}). Without loss of generality it can be assumed that for k_{1} = 0 this mapping is surjective. Fix the integer k_{1} > 0. By definition, put P_{2} = coker φ_{Δ}^{k+k_{1}} and let φ_{Δ_{1}} be the natural projection J̄^{k_{1}}(P_{1}) → P_{2}. It is clear that Δ_{1} ∘ Δ = 0. Further, starting from an integer k_{2} and the operator Δ_{1}, we construct as above an operator Δ_{2}, such that Δ_{2} ∘ Δ_{1} = 0. Continuing this process, we obtain a complex of the form (2.2).

Is this complex formally exact? To settle the question, let us discuss the notion of involutiveness of a C-differential operator. Consider the Spencer δ-complex of the module P_{0} (see Diagram 2.1)

0 ⟶ S̄^{r} ⊗ P_{0} ⟶^{δ̄} Λ̄^{1} ⊗ S̄^{r−1} ⊗ P_{0} ⟶^{δ̄} Λ̄^{2} ⊗ S̄^{r−2} ⊗ P_{0} ⟶^{δ̄} ⋯  (2.5)

Let g^{k+l} ⊂ S̄^{k+l} ⊗ P_{0} be the symbolic module of the operator Δ, i.e., g^{k+l} = ker σ(Δ). It is easily shown that the subcomplex of complex (2.5)

0 ⟶ g^{k+l} ⟶^{δ̄} Λ̄^{1} ⊗ g^{k+l−1} ⟶^{δ̄} Λ̄^{2} ⊗ g^{k+l−2} ⟶^{δ̄} ⋯  (2.6)

is well defined. Cohomology of this complex in the term Λ̄^{i} ⊗ g^{k+l−i} is denoted by H̄^{k+l,i}(Δ) and is said to be horizontal Spencer δ-cohomology of the operator Δ. Note that H̄^{k+l,0}(Δ) = H̄^{k+l,1}(Δ) = 0. The operator Δ is called involutive (in the sense of Cartan), if H̄^{k+l,i}(Δ) = 0 for all i ≥ 0.

Theorem 2.20. If the operator Δ is involutive, then the complex of the form (2.2) constructed as described above is formally exact for all positive integers k_{1}, k_{2}, k_{3}, ….

Proof. We must prove that the sequences

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

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

0  0  0
↑  ↑  ↑
0 ⟶ S̄^{l} ⊗ P_{i+1} ⟶^{δ̄} Λ̄^{1} ⊗ S̄^{l−1} ⊗ P_{i+1} ⟶^{δ̄} Λ̄^{2} ⊗ S̄^{l−2} ⊗ P_{i+1} ⟶^{δ̄} ⋯
↑  ↑  ↑
0 ⟶ S̄^{k_{i}+l} ⊗ P_{i} ⟶^{δ̄} Λ̄^{1} ⊗ S̄^{k_{i}+l−1} ⊗ P_{i} ⟶^{δ̄} Λ̄^{2} ⊗ S̄^{k_{i}+l−2} ⊗ P_{i} ⟶^{δ̄} ⋯
↑  ↑  ↑
⋮  ⋮  ⋮
↑  ↑  ↑
0 ⟶ S̄^{K+l} ⊗ P_{0} ⟶^{δ̄} Λ̄^{1} ⊗ S̄^{K+l−1} ⊗ P_{0} ⟶^{δ̄} Λ̄^{2} ⊗ S̄^{K+l−2} ⊗ P_{0} ⟶^{δ̄} ⋯
↑  ↑  ↑
0 ⟶ g^{K+l}  ⟶^{δ̄}  Λ̄^{1} ⊗ g^{K+l−1}  ⟶^{δ̄}  Λ̄^{2} ⊗ g^{K+l−2}  ⟶^{δ̄} ⋯
↑  ↑  ↑
0  0  0

where K = k + k_{1} + k_{2} + ⋯ + k_{i}.

Definition 2.21. A formally exact complex of the form (2.2) is called the compatibility complex for the operator Δ.

For a discussion of the compatibility complex see [4, 23, 25].

The condition of involutiveness is not necessary for the existence of the compatibility complex. Really, the δ-Poincaré lemma (see, for example, [15, 4]) says that for any C-differential operator Δ there exists an integer l_{0} = l_{0}(m, n, k), where m = rank P, such that H̄^{k+l,i}(Δ) = 0 for l ≥ l_{0} and i ≥ 0. Taking into account this fact, we see from the proof of the previous theorem that for sufficiently large integer k_{1} the compatibility complex exists for any C-differential operator Δ.

The following proposition is obvious.

Proposition 2.22. In the compatibility complex each operator Δ_{i+1} is a compatibility operator for the preceding operator Δ_{i}, i.e., for any operator ∇ and an integer l ≥ 0 such that ∇ ∘ Δ_{i} = 0 and ord ∇ ≥ k_{i+1} − l, there exists an operator □ such that ∇^{(l)} = □ ∘ Δ_{i+1}, where ∇^{(l)} = j̄_{l} ∘ ∇ is the l-th prolongation of ∇.

Example 2.23. The de Rham complex is the compatibility complex for the operator d̄: F → Λ̄^{1}. The proof is trivial.

Example 2.24. Fix an F-linear scalar product of index i on the module Λ̄^{1}. For an integer p ≥ 1 consider the operator Δ = d̄∗d̄: Λ̄^{p} → Λ̄^{n−p}, where ∗: Λ̄^{k} → Λ̄^{n−k} is the Hodge star operator. Let us show that the complex

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

is the compatibility complex for the operator Δ. Indeed, we must prove that the image of the map σ(Δ): S̄^{l+2} ⊗ Λ̄^{p} → S̄^{l} ⊗ Λ̄^{n−p} coincides with the image of the map σ(d̄): S̄^{l+1} ⊗ Λ̄^{n−p−1} → S̄^{l} ⊗ Λ̄^{n−p} for all l ≥ 0. Since Δ∗ = d̄∗ d̄∗ = d̄(∗d̄∗ +(−1)^{pn+n+i}d̄), it is sufficient to show that the map σ(∗d̄∗ +(−1)^{pn+n+i}d̄): S̄^{l+1} ⊗ Λ̄^{n−p−1} → S̄^{l} ⊗ Λ̄^{n−p} is an epimorphism. Take an element ξ ∈ Λ̄^{1}. One has σ(d̄)(ξ^{l+1} ⊗ ω) = lξ^{l} ⊗ ξ ∧ ω. Denote by A_{n−p−1} : Λ̄^{n−p−1} → Λ̄^{n−p} the map of exterior multiplication by ξ. We have A_{n−p}^{∗} = (−1)^{pn+n+i} ∗ A_{p−1}∗, so it will suffice to check that the map A_{n−p−1} + A_{n−p}^{∗} is an epimorphism. But this is obvious: Λ̄^{n−p} = im A_{n−p−1} ⊕ (im A_{n−p−1})^{⊥} = im A_{n−p−1} ⊕ (ker A_{n−p})^{⊥} = im A_{n−p−1} ⊕ im A_{n−p}^{∗} = im(A_{n−p−1} + A_{n−p}^{∗})^{1}.

3. Applications to computing cohomological invariants of systems of differential equations

3.1. Main theorems. Let E = {F = 0}, F ∈ P_{1}, be an equation,

P_{0} = κ ⟶^{ℓ_{F}} 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, and

P̂_{0} = κ̂ ⟵^{ℓ_{F}^{∗}} P̂_{1} ⟵^{Δ_{1}^{∗}} P̂_{2} ⟵^{Δ_{2}^{∗}} P̂_{3} ⟵^{Δ_{3}^{∗}} P̂_{4} ⟵^{Δ_{4}^{∗}} ⋯

the dual complex. Take a C-module Q.

Theorem 3.1. H̄^{i}(D^{v}(Q)) = H^{i}(P_{•} ⊗ Q), H̄^{n−i}(CΛ^{1} ⊗ Q) = H_{i}(P̂_{•} ⊗ Q).

Proof. The statement follows immediately from (2.3), (2.4) and Proposition 1.1.

Let Q = C^{p}Λ^{p}. The previous theorem gives, first, a method for computing of the cohomology groups H̄^{i}(D^{v}(C^{p}Λ^{p})), which are Krasil′shchik’s C-cohomol-ogy groups (see Example 2.18). Second, since the term E_{1}^{p,q} = H̄^{q}(C^{p}Λ^{p}) of the Vinogradov C-spectral sequence is a direct summand in the cohomology group H̄^{q}(CΛ^{1} ⊗ C^{p−1}Λ^{p−1}), we have a description for the first term of the Vinogradov C-spectral sequence. Thus:

Corollary 3.2. H̄^{i}(D^{v}(C^{p}Λ^{p})) = H^{i}(P_{•} ⊗ C^{p}Λ^{p}).

Corollary 3.3. The term E_{1}^{p,q} of the Vinogradov C-spectral sequence is the skewsymmetric part of the group H_{n−q}(P̂_{•} ⊗ C^{p−1}Λ^{p−1}).

It is useful to describe the isomorphisms given by these corollaries in an explicit form. We discuss the Vinogradov C-spectral sequence, the case of Krasil′shchik’s C-cohomology is similar.

Consider an operator ∇ ∈ CDiff(κ, Λ̄^{q} ⊗ C^{p−1}Λ^{p−1}) that represents an element of E_{1}^{p,q}. This means that

d̄∘ ∇ = ∇_{1} ∘ ℓ_{F}

for an operator ∇_{1} ∈ CDiff(P_{1}, Λ̄^{q+1} ⊗C^{p−1}Λ^{p−1}). Applying the operator d̄to both sides of this formula and using Proposition 2.22, we get

d̄∘ ∇_{1} = ∇_{2} ∘ Δ_{1}

for an operator ∇_{2} ∈ CDiff(P_{2}, Λ̄^{q+2} ⊗ C^{p−1}Λ^{p−1}). Continuing this process, we obtain operators ∇_{i} ∈ CDiff(P_{i}, Λ̄^{q+i} ⊗ C^{p−1}Λ^{p−1}), 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^{p−1}) represents an element of the module P̂_{n−q} ⊗ C^{p−1}Λ^{p−1} 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^{p−1}) corresponding to the chosen element of E_{1}^{p,q}.

1I thank D. Gessler for drowning my attention to this example (cf. [6]).

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_{F}} P_{1} ⟶^{Δ_{1}} P_{2} ⟶^{Δ_{2}} P_{3} ⟶^{Δ_{3}} ⋯ ⟶^{Δ_{k−2}} P_{k−1} ⟶ 0,

then 1. E_{1}^{p,q} = 0 for p > 0 and q ≤ n − k,

2. H^{i}(D^{v}(C^{p}Λ^{p})) = 0 for i ≥ k. This result is known as the k-line theorem.

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. The majority of systems that occur in practice consist of independent equations and for them k = 2. In this case the two-line theorem holds:

Theorem 3.4 (the two-line theorem). Let an equation E be such that the compatibility complex for ℓ_{F} is of length two. Then:

1. E_{1}^{p,q} = 0 for p > 0 and q ≤ n − 2,

2. E_{1}^{p,n−1} ⊂ ker(ℓ_{F}^{∗})_{C^{p−1}Λ^{p−1}} for p > 0,

3. E_{1}^{p,n} ⊂ coker(ℓ_{F}^{∗})_{C^{p−1}Λ^{p−1}} for p > 0,

4. H^{i}(D^{v}(C^{p}Λ^{p})) = 0 for i ≥ 2,

5. H^{0}(D^{v}(C^{p}Λ^{p})) = ker(ℓ_{F})_{C^{p}Λ^{p}},

6. H^{1}(D^{v}(C^{p}Λ^{p})) = coker(ℓ_{F})_{C^{p}Λ^{p}}.

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. 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.

3.2. 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_{F} = D_{t} − ℓ_{f} = D_{t} − ∑_{i} \frac{∂f}{∂u_{i}} D_{x}^{i}.

The adjoint of ℓ_{F} is

ℓ_{F}^{∗} = −D_{t} − ℓ_{f}^{∗} = D_{t} − ∑_{i}(−1)^{i}D_{x}^{i} ∘ \frac{∂f}{∂u_{i}} .

Clearly, for an evolution equation the two-line theorem holds, hence Krasil′- shchik’s C-cohomology H̄^{q}(D^{v}(C^{p}Λ^{p})) is trivial for q ≥ 2 and the first term E_{1}^{p,q} of the Vinogradov C-spectral sequence is trivial for q ≠ 1, 2 and p > 0. 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 3.5. For any evolution equation of order ≥ 2, one has 1. H̄^{0}(D^{v}(C^{p}Λ^{p})) = 0 for p ≥ 2,

2. E_{1}^{p,1} = 0 for p ≥ 3.

Proof. It follows from Theorem 3.4 that H̄^{0}(D^{v}(C^{p}Λ^{p})) = ker(ℓ_{F})_{C^{p}Λ^{p}} and E_{1}^{p,1} = ker(ℓ_{F}^{∗})_{C^{p−1}Λ^{p−1}}. Hence to prove the theorem it suffices to check that equations

(D_{t} − ℓ_{f})(ω) = 0 and (D_{t} + ℓ_{f}^{∗})(ω) = 0,  (3.1)

with ω ∈ C^{p}Λ^{p}, has no nontrivial solutions for p ≥ 2.

To this end, consider the symbol of (3.1). Denote σ(D_{x}) = θ. The symbol of ℓ_{F} has the form σ(ℓ_{f}) = fθ^{k}, k ≥ 2. An element ω ∈ C^{p}Λ^{p} can be identified with a multilinear C-differential operator, so the symbol of ω is a homogeneous polynomial in p variables σ(ω) = δ(θ_{1}, …, θ_{p}). Either of the two equations (3.1) yields

[f(θ_{1}^{k} + ⋯ + θ_{p}^{k}) ± f(θ_{1} + ⋯ + θ_{p})^{k}](δ) = 0.

The conditions k ≥ 2 and p ≥ 2 obviously imply that δ = 0. This completes the proof.

Remark 3.6. In the work [7] this proof has been generalized for determined systems of evolution equations with arbitrary number of independent variables.

3.3. Example: Abelian p-form theories. Let M be a (pseudo-)Riemannian 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. Our aim is to calculate the terms of the Vinogradov C-spectral sequence E_{1}^{i,q} 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_{F} = d̄∗d̄: Λ̄^{p} → Λ̄^{n−p}. Taking into account Example 2.24, we see that the compatibility complex for ℓ_{F} has the form

\begin{matrix}Λ̄^{p} & ⟶^{ℓ_{F}} & Λ̄^{n−p} & ⟶^{d̄} & Λ̄^{n−p+1} & ⟶^{d̄} & ⋯ & ⟶^{d̄} & Λ̄^{n} & ⟶ & 0 \\ ∥ & & ∥ & & ∥ & & & & ∥ \\ P_{0} & & P_{1} & & P_{2} & & & & P_{k−1} \end{matrix} (3.2)

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 3.3 one has

E_{1}^{i,q} ⊂ H̄^{q−(n−p−1)}(C^{i−1}Λ^{i−1}) = E_{1}^{i−1,q−(n−p−1)}

because the complex dual to the compatibility complex (3.2) has the form

Λ̄^{n−p} ⟵^{ℓ_{F}^{∗}} 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

[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 3.1

that the one-dimensional space E_{1}^{0,n−p−1} is 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 3.1).

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 skewsymmetric, 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 3.7. For i = q = 0 one has dim E_{1}^{0,0} = 1. If either or both i or 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} ⊗ CΛ^{1}; 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. In particular, for horizontal cohomology with trivial coefficients this result agrees with that of [8] obtained previously by means of the Koszul - Tate resolution.

Remark 3.8. One can consider the quotient equation under the action of gauge symmetries. The techniques described here allow one to compute the terms E_{1}^{i,q}, q < n − 1, for the quotient equation. This will be considered elsewhere.

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Moscow State Technical University and The Diffiety Institute, Moscow, Russia. Correspondence to: A. M. Verbovetsky, Profsoyuznaya 98-9-132, 117485 Moscow, Russia

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