The Diffiety Institute Preprint Series

Preprint DIPS–6/2001 May 17, 2001

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

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

Remarks on two approaches to the

horizontal cohomology: compatibility complex and the

Koszul-Tate resolution

by

Alexander Verbovetsky

Preprint ESI 1032

To appear in Acta Appl. Math.

Remarks on two approaches to the horizontal cohomology: compatibility complex and the Koszul-Tate resolution

ALEXANDER VERBOVETSKY

Abstract. The Koszul-Tate resolution is described in the context of the geometry of jet spaces and differential equations. The application due to Barnich, Brandt, and Henneaux of this resolution to computing the horizontal cohomology is analyzed. Relations with the Vinogradov spectral sequence are discussed.

The Perl motto is “There’s more than one way to do it.” Divining how many more is left as an exercise to the reader.

Larry Wall, The Perl man page

1. Introduction

This paper is concerned with general methods for computing horizontal (also called “characteristic”) cohomology of systems of nonlinear partial differential equations. There are at least two such methods. One stems from the fact that the horizontal cohomology is the column E_{1}^{0,•} of the Vinogradov spectral sequence [8, 9, 10] and thereby related to the terms E_{1}^{p,•} for p > 0. These terms can be computed via the compatibility complex for the linearization of the system under consideration. For a detailed description of this technique the reader should consult [6, 7].

The second method was proposed in [1] and is based on the Koszul-Tate resolution [4]. The purpose of the present paper is to describe this method in the language of the geometry of differential equations (see, e.g., [2, 5]) and look into relationships between this method and the former one.

We restrict the discussion to a general theory. As an example we refer to the p-form gauge theory that was explicitly worked out in [3] by means of the Koszul-Tate resolution and in [7, 6] by means of the Vinogradov spectral sequence.

Date: May 2001.

2000 Mathematics Subject Classification. 58J70, 35A30, 81T70.

Key words and phrases. Jet space, Partial differential equation, Horizontal cohomology, Vinogradov spectral sequence, Compatibility complex, the Koszul-Tate resolution.

Research supported in part by the GNFM of the INdAM (Italy), the University of Lecce, the Italian MURST Young Researchers Project, and the ESI (Vienna).

2. On compatibility operators

We assume that the reader is familiar with the geometry of jet spaces and differential equations, including the horizontal cohomology, the Vinogradov spectral sequence, the compatibility complex, and the k-line theorem. This material can be found in [6, 7].

Let π: E → M be a vector bundle and π_{∞}: J^{∞}(π) → M the associated infinite jet bundle. The standard coordinates on the space J^{∞}(π) are coordinates x_{i} of the manifold M and infinitely many coordinates u_{σ}^{j}, corresponding to partial derivatives of sections u = (u^{1}, …, u^{m}) of the bundle π.

Remark. All constructions of the present paper can be readily generalized to the case of vector bundle π with super fibers.

Let P_{1} = Γ(π_{∞}^{∗}(α_{1})) be a horizontal module on J^{∞}(π) (as usual, a module is always the module of sections of a vector bundle; Γ denotes the functor that takes bundles to the corresponding modules), E = {F = 0} ⊂ J^{k}(π) a formally integrable equation, determined by a section F ∈ P_{1}, and E^{∞} ⊂ J^{∞}(π) its infinite prolongation.

Remark. Physicists called coordinates along fibers of α_{1} antifields and say that they have the antighost number 1 (hence the notation P_{1}).

Regularity Condition ([4]). We shall assume that there exists an open submanifold U ⊂ J^{∞}(π) and an isomorphism ν: U → E^{∞}×V, where V is a starshaped neighborhood of the zero in ℝ^{∞}, such that U ⊃ E^{∞}, ν(θ) = (θ, 0) for any point θ ∈ E^{∞}, and the composition v: U ⟶^{ν} E^{∞} × V → V of the isomorphism ν and the projection on the second factor has the form

v = (v_{1}, v_{2}, v_{3}, …, v_{s}, …),

where v_{s} = D_{σ_{s}}(F_{i_{s}}).

Because of the regularity condition, a function f on U vanishes on E^{∞} if and only if it has the form f = Δ(F) for some operator Δ ∈ CDiff(P_{1}, F), where F is the algebra of functions on U.

From here on, we fix a manifold U and will consider it instead of the whole jet space J^{∞}(π).

Lemma 1. Let Q = Γ(π_{∞}^{∗}(α)) be a horizontal module; then the kernel of the map

Q ⟶ CDiff(κ|_{E^{∞}} , Q|_{E^{∞}}), q ↦ ℓ_{q}|_{E^{∞}} ,

where κ = Γ(π_{∞}^{∗}(π)) and ℓ_{q} is the linearization of q, has the form I^{2}Q⊕Q′ ⊂ Q, where I = { Δ(F) | CDiff(P_{1}, F) } ⊂ F is the ideal of the equation E^{∞} and Q′ = { π_{∞}^{∗}(s) | s ∈ Γ(α) }. (In other words, the set I^{2}Q ⊂ Q consists of elements of the form Δ(F, F) for Δ ∈ CDiff(P_{1}, CDiff(P_{1}, Q)).)

Proof. It suffices to consider the case Q = F and to prove the statement in a local chart. Take a function f ∈ F such that ℓ_{f} = 0. Then ∂f/∂u_{σ}^{j} ∈ I. Now let us change coordinates to (x_{i}, v_{s}, w_{t}), where v_{s} are the coordinates along V from the regularity condition and w_{t} are arbitrary coordinates along E^{∞}. We have ∂f/∂v_{s} ∈ I and ∂f/∂w_{t} ∈ I. Write the function f in the form

f = f_{0}(x_{i}, w_{t}) + ∑_{s} f_{s}(x_{i}, w_{t})v_{s} + f′,

where f′ ∈ I^{2}. Then ∂f/∂v_{s} = f_{s} + ∂f′/∂v_{s} ∈ I, hence f_{s} = 0, and ∂f/∂w_{t} = ∂f_{0}/∂w_{t} + ∂f′/∂w_{t} ∈ I, so that ∂f/∂w_{t} = 0. Thus, f ∈ I^{2} ⊕ π_{∞}^{∗}(C^{∞}(M)). □

Proposition 2. An operator Δ ∈ CDiff(P_{1}|_{E^{∞}} , P_{2}|_{E^{∞}}) is the compatibility operator for ℓ_{F} |_{E^{∞}} if and only if for each operator ∇: P_{1} → Q such that ∇(F) = 0 we have ∇|_{E^{∞}} = □ ∘ Δ for an operator □ ∈ CDiff(P_{2}|_{E^{∞}} , Q|_{E^{∞}}).

Proof. Suppose that Δ ∈ CDiff(P_{1}|_{E^{∞}} , P_{2}|_{E^{∞}}) is the compatibility operator for ℓ_{F} |_{E^{∞}}. Linearizing the equality ∇(F) = 0, we get ∇|_{E^{∞}} ∘ ℓ_{F} |_{E^{∞}} = 0. Hence, ∇|_{E^{∞}} = □ ∘ Δ.

Conversely, consider an operator ∇ ∈ CDiff(P_{1}, Q) satisfying the condition ∇|_{E^{∞}} ∘ ℓ_{F} |_{E^{∞}} = 0. Since ℓ_{∇(F)}|_{E^{∞}} = ∇|_{E^{∞}} ∘ ℓ_{F} |_{E^{∞}} = 0, we see that ∇(F) = ∇′(F, F). Define the operator ∇_{1} ∈ CDiff(P_{1}, Q) by the equality ∇_{1}(p) = ∇(p) − ∇′(F, p). We have ∇_{1}(F) = 0, so that ∇|_{E^{∞}} = ∇_{1}|_{E^{∞}} = □ ∘ Δ. This completes the proof. □

An equation is called normal if the compatibility operator for the operator ℓ_{F} |_{E^{∞}} is trivial.

Corollary 3. The equation E^{∞} is normal if and only if for each operator ∇: P_{1} → Q the equality ∇(F) = 0 implies ∇|_{E^{∞}} = 0.

Corollary 4 (from the proof). Suppose that Δ ∈ CDiff(P_{1}|_{E^{∞}} , P_{2}|_{E^{∞}}) is the compatibility operator for ℓ_{F} |_{E^{∞}}; then there exist an extension Δ̃ ∈ CDiff(P_{1}, P_{2}) of Δ such that Δ̃(F) = 0.

3. The Koszul-Tate resolution

Due to the last Corollary there is an operator Δ_{1} : P_{1} → P_{2} such that Δ_{1}|_{E^{∞}} is the compatibility operator for ℓ_{F} |_{E^{∞}} and Δ_{1}(F) = 0. Pick up such an operator and construct the compatibility complex for it:

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

where P_{i} = Γ(α_{i}) for some vector bundles α_{i}: V_{i} → U.

Remark. Such a complex may not exist (modules P_{i} may not be projective). In [4] the assumption that it exists is termed as “off-shell reducibility.” The situation then this complex exists only on E^{∞} is called “on-shell reducibility.” So, we here require the off-shell reducibility.

Take the direct sum α = ⨁_{i≥1}(α_{2i−1}^{Π} ⊕ α_{2i}), where α_{2i−1}^{Π} means the bundle α_{2i−1} with reversed parity of fibers, and consider the supermanifold of horizontal jets of its sections J̄^{∞}(α).

Each Cartan field on U can be naturally lifted to J̄^{∞}(α); these liftings span the Cartan distribution on the horizontal jets J̄^{∞}(α). It is not hard to check that all the theory of jet spaces can be carried over to the space J̄^{∞}(α). In coordinate language, the horizontal jet space is a jet space equipped with extra base coordinates u_{σ}^{j} (so that all functions depend on them as on parameters), with the total derivatives D_{i} = ∂/∂x_{i} +∑_{j,σ} u_{σi}^{j}∂/∂u_{σ}^{j} in place of the partial derivatives ∂/∂x_{i}.

Now let us pull the element F and all operators Δ_{i} back to J̄^{∞}(α). We shall treat them as elements of the module κ(α) = Γ(α^{∗}(α)). Consider the element Φ = F + Δ_{1} + ⋯ + Δ_{k−2} ∈ κ(α).

Proposition 5. The odd evolutionary vector field δ = �_{Φ} is a differential: δ^{2} = 0.

Proof. Since δ is a vector field and δ|_{F} = 0, it is sufficient to evaluate δ^{2} on functions linear along the fibers of the natural projection α_{∞}: J̄^{∞}(α) → U. Such functions can be naturally identified with C-differential operators belonging to CDiff(P, F), where P = Γ(α) = ⨁_{i≥1}(Γ(α_{2i−1}^{Π}) ⊕ Γ(α_{2i})) is a graded F-module. Define an odd C-differential operator Δ: P → P by the formula

Δ(p_{1}, …, p_{k−2}, p_{k−1}) = (0, Δ_{1}(p_{1}), …, Δ_{k−2}(p_{k−2})),

where p_{i} ∈ P_{i} = Γ(α_{i}). Trivially, Δ^{2} = 0 and Δ(F) = 0. It is easy to see that if ∇ ∈ CDiff(P, F) then δ(∇) = ∇(F) + ∇ ∘ Δ, so that δ^{2}(∇) = ∇ ∘ Δ^{2} = 0. □

Denote by F^{pol}(α) the subalgebra of the algebra F(α) of functions on J̄^{∞}(α) that consists of functions polynomial along the fibers of the projection α_{∞}: J̄^{∞}(α) → U. We supply the algebra F^{pol}(α) with a ℤ-grading, F^{pol}(α) = ⨁_{i≥0} F_{i}^{pol}(α), called the antighost number, such that fiberwise linear functions on J̄^{∞}(α_{i}) have antighost number i and functions on U have antighost number zero. (Thus the parity of a function is equal to its antighost number modulo 2.) The differential δ reduces the antighost number by 1 and so we have the complex

0 ⟵ F ⟵^{δ} F_{1}^{pol}(α) ⟵^{δ} F_{2}^{pol}(α) ⟵^{δ} ⋯

called the Koszul-Tate complex [4].

Theorem 6. The Koszul-Tate complex is a resolution, with the zero homology isomorphic to the algebra F(E) of functions on the equations E^{∞}. The homology of the differential group (F(α), δ) is equal to F(E) as well.

Proof. See [4]. □

Let us consider the bicomplex (F_{p}^{pol}(α) ⊗ Λ̄^{q}, δ, (−1)^{q}d̄), where Λ̄^{q} is the module of horizontal q-forms on U. The horizontal differential d̄ is welldefined, since there is a natural inclusion F_{p}^{pol}(α) ⊗ Λ̄^{q} ⊂ Λ̄^{q}(α), where Λ̄^{q}(α) is the module of horizontal q-forms on J̄^{∞}(α). Using this bicomplex and Theorem 6, we immediately conclude that for 0 ≤ q ≤ n − 1 the group H̄^{q}(E)/H̄^{q}(U) is isomorphic to the (n − q)^{th} homology group of the complex

H̄_{0}^{n}(α) ⟵^{δ} H̄_{1}^{n}(α) ⟵^{δ} H̄_{2}^{n}(α) ⟵^{δ} ⋯,  (1)

where H̄_{i}^{n}(α) = H^{n}(F_{i}^{pol}(α) ⊗ Λ̄^{q}, d̄). Application of the one-line theorem for U yields the following reformulation of this result.

Theorem 7. For 0 ≤ q ≤ n − 1 there is the isomorphism

H̄^{q}(E)/H^{q}(M) = H_{n−q}(H̄_{•}^{n}(α), δ).

Our main concern now is to compute the homology of complex (1). To this end it is convenient to drop the zero-term H̄_{0}^{n}(α) and to embed the rest of the complex to the complex (κ̂(α), δ +ℓ_{Φ}^{∗}) by means of the Euler operator

⨁_{i≥1} H̄_{i}^{n}(α) → κ̂(α)  (2)

on J̄^{∞}(α).

Remark. We use here two facts:

(1) the map δ + ℓ_{Φ}^{∗}: κ̂(α) → κ̂(α) is a differential: (δ + ℓ_{Φ}^{∗})^{2} = 0;

(2) the Euler operator (2) is a chain map. Both of these statements follow from the infinitesimal Stokes formula in the term E_{1} of the Vinogradov spectral sequence on J̄^{∞}(α) (see [10, Sec. 9.10]^{1}).

Proposition 8. The operator ℓ_{Φ}^{∗}: κ̂(α) → κ̂(α) is a differential, i.e., (ℓ_{Φ}^{∗})^{2} = 0.

Proof. Pick up an element θ ∈ κ̂(α). It can be thought of as a nonlinear operator θ: P → P̂. Obviously, ℓ_{Φ}^{∗}(θ) = (−1)^{p(θ)}Δ^{∗} ∘ θ, where p(θ) is the parity of θ. Hence, (ℓ_{Φ}^{∗})^{2}(θ) = −(Δ^{∗})^{2} ∘ θ = 0. □

Remark. Note that the operator Δ^{∗}: P̂ → P̂ has the form

Δ^{∗}(p̂_{1}, p̂_{2}, …, p̂_{k−1}) = (−Δ_{1}^{∗}(p̂_{2}), Δ_{2}^{∗}(p̂_{3}), …, (−1)^{k−2}Δ_{k−2}^{∗}(p̂_{k−1})),

where p̂_{i} ∈ P̂_{i}.

Thus, we see that the differentials δ and ℓ_{Φ}^{∗} define a bicomplex structure in κ̂(α). The same holds true for the polynomial part κ̂^{pol}(α) = F^{pol}(α)⊗P̂. This enables an easy computation of the homology of the total complex (κ̂(α), δ + ℓ_{Φ}^{∗}) and its polynomial subcomplex.

χ{φ} = l_{φ}(χ) + l_{χ}^{∗}(φ) = �_{χ}(φ) + l_{χ}^{∗}(φ).

1Note a misprint in [10, eq. (9.10.5)]. The correct formula (in the notation of [10]) is

Proposition 9. The homologies of complexes

(κ̂(α), δ + ℓ_{Φ}^{∗})  and  (κ̂^{pol}(α), δ + ℓ_{Φ}^{∗})

coincide and equal the homology of the complex

0 ⟵ P̂_{1} ⟵^{Δ_{1}^{∗}} P̂_{2} ⟵^{Δ_{2}^{∗}} ⋯ ⟵^{Δ_{k−2}^{∗}} P̂_{k−1} ⟵ 0  on the equation E^{∞}. (3)

Corollary 10. Each cycle ψ belonging to κ̂_{+}^{pol}(α) = (⨁_{i≥1}^{∞} F_{i}^{pol}(α)) ⊗ P̂ is a boundary.

From the one-line theorem for J̄^{∞}(α) it follows that the image of the mapping (2) coincides with the complex (κ̂^{sp}(α), δ + ℓ_{Φ}^{∗}), where

κ̂^{sp}(α) = { ψ ∈ κ̂^{pol}(α) | ℓ_{ψ}^{∗} = ℓ_{ψ} }.

Remark. It should be noted that κ̂^{sp}(α) does not inherit the bicomplex structure of κ̂^{pol}(α).

The space κ̂^{sp}(α) can be expanded in the sum

κ̂^{sp}(α) = P̂ ⊕ κ̂_{+}^{sp}(α),

where κ̂_{+}^{sp}(α) = κ̂^{sp}(α) ∩ κ̂_{+}^{pol}(α).

Proposition 11. Each cycle ψ belonging to κ̂_{+}^{sp}(α) is a boundary in the complex (κ̂^{sp}(α), δ + ℓ_{Φ}^{∗}).

Proof. We prove the statement by identifying in a natural way elements ψ ∈ κ̂^{pol}(α) with graded symmetric multilinear C-differential operators

∇_{ψ}: P × ⋯ × P → P̂.

It is not hard to check that ψ ∈ κ̂^{sp}(α) if and only if the corresponding operator ∇_{ψ} is selfadjoint: ∇_{ψ} = ∇_{ψ}^{∗} (since ∇_{ψ} is symmetric, self-adjointness in one argument implies self-adjointness in the other arguments). Consider the projector S: κ̂^{pol}(α) → κ̂^{sp}(α) given by S(∇) = (∇ + ∇^{∗})/2. Obviously, S^{2} = S and S ∘ (δ + ℓ_{Φ}^{∗}) = (δ + ℓ_{Φ}^{∗}) ∘ S. Now, if a cycle ∇ belongs to κ̂_{+}^{sp}(α) then by Corollary 10 we have ∇ = (δ + ℓ_{Φ}^{∗})∇′. This gives ∇ = S(∇) = S(δ + ℓ_{Φ}^{∗})∇′ = (δ + ℓ_{Φ}^{∗})(S(∇′)), which is the desired conclusion. □

Corollary 12. Complex (1) is exact in terms H̄_{i}^{n}(α) for i ≥ k.

Corollary 13. If 0 ≤ q ≤ n − k then H̄^{q}(E) = H^{q}(M).

Take a cycle ψ ∈ κ̂^{sp}(α) and write it in the form ψ = ψ_{0} + ψ_{1}, where ψ_{0} ∈ P̂ and ψ_{1} ∈ κ̂_{+}^{sp}(α). Obviously, ψ_{0}|_{E^{∞}} is a cycle of complex (3). So, we have constructed a map from homology of complex (1) to homology of complex (3).

Let us examine this map more fully in dimension 1. We have H̄_{1}^{n}(α) = P̂_{1} and the map being considered is just the restriction P̂_{1} → P̂_{1}|_{E^{∞}}. Further, an element β ∈ H̄_{1}^{n}(α) = P̂_{1} is a cycle if and only if the horizontal cohomology class of the n-form ⟨β, F ⟩ is trivial, so that 0 = ℓ_{⟨β,F ⟩}^{∗}(1) = ℓ_{β}^{∗}(F)+ℓ_{F}^{∗} (β) and, hence, ℓ_{F}^{∗} |_{E^{∞}} (β|_{E^{∞}}) = 0. Thus, we get a map from homology of complex (1) to homology of the complex

κ ⟵^{ℓ_{F}^{∗}} P̂_{1} ⟵^{Δ_{1}^{∗}} P̂_{2} ⟵^{Δ_{2}^{∗}} ⋯ ⟵^{Δ_{k−2}^{∗}} P̂_{k−1} ⟵ 0  on the equation E^{∞}. (4)

The application of Theorem 7 yields the map

H̄^{q}(E) → H_{n−q}(P̂_{•}, Δ_{•}^{∗}).  (5)

It is straightforward to check that this map coincides with the differential d_{1}^{0,q}: E_{1}^{0,q} = H̄^{q}(E) → E_{1}^{1,q} = H_{n−q}(P̂_{•}, Δ_{•}^{∗}) of the Vinogradov spectral sequence on E^{∞}.

4. A comparison

As it is seen from the last paragraphs of the previous section, the homology of complex (4) is essential to computing the horizontal cohomology via the Koszul-Tate resolution, similarly to what happens when using the Vinogradov spectral sequence. This bridges the gap between two approaches. Otherwise they are diverged considerably.

As an example, let us discuss the computation of conservation laws for a normal equation (i.e., such that Δ_{1} = 0). The application of the Koszul-Tate resolution gives the following result.

Proposition 14. Let E^{∞} be a normal equation. The space of conservation laws of E^{∞} is isomorphic to the space of solutions of the equation

ℓ_{F}^{∗} (ψ) + ℓ_{ψ}^{∗}(F) = 0,

where ψ ∈ P̂_{1}/Θ and Θ = { □(F) ∈ P̂_{1} | □ ∈ CDiff(P_{1}, P̂_{1}), □ = −□^{∗} }.

Another result is obtained by means of the Vinogradov spectral sequence.

Proposition 15. Let E^{∞} be a normal equation. The space of conservation laws of E^{∞} is a subset of the space of solutions of the equation

ℓ_{F}^{∗} (ψ) = 0  on E^{∞},  (6)

where ψ ∈ P̂_{1}|_{E^{∞}}. A solution ψ corresponds to a conservation law if and only if on E^{∞} there exists a selfadjoint C-differential operator ∇: P_{1} → P̂_{1} such that

ℓ_{ψ} + Δ^{∗} = ∇ ∘ ℓ_{F} , ∇^{∗} = ∇ on E^{∞},

where Δ: P_{1} → κ̂ is a C-differential operator satisfying on U the equality ℓ_{F}^{∗} (ψ) = Δ(F).

Thus, both of these Propositions say that to compute conservation laws of a normal equation we should start with solving equation (6). Proposition 15 also says that if ψ vanishes on E^{∞} then the corresponding conservation law is trivial (essentially, this is the basic content of the Proposition). Proposition 14 implies a weaker result: it guaranteers triviality of conservation laws that correspond to elements ψ of the form ψ = □(F) for skew-adjoint operators □ only.

Acknowledgements

The author wishes to extend his thanks to F. Brandt, I. S. Krasil′shchik, and A. M. Vinogradov for many helpful discussions. He is also grateful to the Erwin Schrödinger Institute in Vienna, where a part of this research was completed, for kind hospitality.

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Independent University of Moscow. Correspondence to: A. Verbovetsky, Profsoyuznaya 98-9-132, 117485 Moscow, Russia.

E-mail address: verbovet@mccme.ru