Characteristic Cohomology of Differential systems (I):

General Theory

by Robert L. Bryant and Phillip A. Griffiths

TABLE OF CONTENTS

Introduction

1. Basic Considerations and First examples 1.1 Definition of characteristic cohomology 1.2 The exterior differential system associated to a partial differential

equation; symmetries 1.3 The prolongation tower 1.4 Variation of characteristic cohomology

2. Frobenius Extensions 2.1 Structure equations of the prolongation tower 2.2 The symbol relations 2.3 Frobenius extensions 2.4 The weight filtration 2.5 A variant of Spencer cohomology

3. Δ-Multilinear and Commutative Algebra 3.1 Poincaré polynomials and Spencer cohomology 3.2 Δ-tensor and exterior products

4. Vanishing of the Characteristic Cohomology 4.1 Characteristic cohomology in the unconstrained case 4.2 Involutivity and Δ-exterior powers 4.3 The tableau and its normal form 4.4 The determined case 4.5 The unmixed, overdetermined case 4.6 The mixed case

5. Structure of the characteristic cohomology 5.1 A variant of the Spencer complex

5.2 Quadratic conservation laws for constant coefficient linear equations

6. Global Results 6.1 A relation between ordinary and characteristic cohomology 6.2 ℐ-homology 6.3 Moment conditions

References Introduction

In this first of a series of papers, we shall introduce and begin to study and apply the notion of the characteristic cohomology of an exterior differential system. Recall that an exterior differential system is a graded, differentially closed ideal ℐ_{0} ⊂ Ω^{∗}(X_{0}) on a manifold X_{0}. It is natural to consider the cohomology of the quotient complex Ω̄_{0}^{∗} = Ω^{∗}(X_{0})/ℐ_{0}. However, this cohomology turns out not to have good functorial properties. For this and other reasons to be explained below, we shall usually pass to the infinite prolongation (X, ℐ) of (X_{0}, ℐ_{0}), and define the characteristic cohomology H̄^{∗} of (X_{0}, ℐ_{0}) to be the cohomology of the quotient complex Ω̄^{∗} = Ω^{∗}(X)/ℐ.

Our first general result is that, in the local involutive case,

(1)  H̄^{q} = 0 for 0 < q < n − ℓ,

where n is the dimension of the maximal integral manifolds (solutions) of the exterior differential system and ℓ is an easily computed integer which measures the “degree of overdeterminedness” of ℐ_{0}. (When ℐ_{0} is “unmixed”—i.e., roughly speaking, it is not composed of several exterior differential systems of different degress of complexity—ℓ is the codimension of the characteristic variety.) The first non-zero group H̄^{n−ℓ} turns out to have a structure not immediately apparent from its definition. For reasons to be discussed below, we define the space 𝒞 of conservation laws of the exterior differential system to be H̄^{n−ℓ}. As we shall see, in all of the cases of exterior differential systems which model classical partial differential equations, this space turns out to be isomorphic to the space of conservation laws as they are generally understood.

For exterior differential systems arising from a system of partial differential equations, this result builds on and generalizes extensive previous work (see the references below). However, a major difference in our viewpoint is that, throughout, we insist on full contact invariance of the constructions. Thus, PDEs which appear to be quite different (such as u_{xx}u_{yy} −u_{xy}^{2} = 1 and u_{xx} −u_{yy} = 0) may still be contact equivalent and therefore have the same characteristic cohomology. Since we are not carrying along unnecessary additional structure, such as that needed to keep track of point invariance, the theory perhaps has greater conceptual simplicity.

0.1.1

We are especially interested in developing methods for effectively computing 𝒞 in examples, and in Part II we begin this task. We are especially interested in exterior differential systems that arise from questions in geometry; for example, the study of gradient flows of functionals defined on classes of immersions which themselves are subject to differential constraints. Now, such exterior differential systems may always be written in local coordinates as PDE systems, but there is generally no preferred coordinate system and the introduction of arbitrarily chosen coordinates frequently obscures the geometry. Even when there are more-or-less natural coordinates, the computation of the conservation laws of a given system has generally relied either on symmetry considerations or inspired guesswork. One of our goals is to provide a complementary, in some cases more systematic, method.

Our proof of the general result (1) has essentially two steps. The first is to consider the spectral sequence of the filtered complex F^{p}Ω^{∗}, where

F^{p}Ω^{∗} = image \{\underbrace{ℐ ∧ … ∧ ℐ}_{p times} ∧ Ω^{∗}(X) → Ω^{∗}(X)\}

This spectral sequence abuts to H^{∗}(X) and the characteristic cohomology is the term E_{1}^{0, ∗}. As will now be explained, invariants of the exterior differential system, such as its symbol, appear naturally in calculation of the terms E_{1}^{p,∗} for p > 0. Using information about these higher degree groups and standard spectral sequence machinery will then allow us to “solve” for the characteristic cohomology in this spectral sequence.

The study of the terms E_{1}^{p, ∗} for fixed p > 0 involves a construction from exterior differential algebra that we shall call Frobenius extension. One thinks of a completely integrable Pfaffian system as being the simplest type of exterior differential system, and Frobenius extensions can be thought of as an attempt to close up a given Pfaffian system relative to the Frobenius integrability condition by adjoining new 1-forms. The actual definition of Frobenius extension is given in Section 2.3, as well as that of a Frobenius tower, which is an iteration of Frobenius extensions. The importance of this concept is that the prolongation tower of an involutive exterior differential system is captured algebraically by the notion of a Frobenius tower. For p > 0 the “position” of a form in the

0.1.2 Frobenius tower allows us to define a weight filtration W_{k}Ω̄^{p, ∗} on the complexes used to compute E_{1}^{p, ∗}, and then the main observation is that on the associated graded complex

Ω̄_{k}^{p, ∗} = W_{k}Ω̄^{p, ∗}/W_{k−1}Ω̄^{p, ∗}

the induced differential δ is linear over the functions. The cohomology

(2)  H(Ω̄_{k}^{p, ∗}, δ), p > 0,

is therefore algebraic and it is at this point that the symbol of the exterior differential system makes its appearance. For p = 1 the cohomology (2) turns out to be a variant of classical Spencer cohomology, but for p ≧ 2 a new algebraic object Λ_{Δ}^{p}M appears.

The second step then consists of the study of the Λ_{Δ}^{p}M. Over a polynomial ring S = F [x^{1}, …, x^{n}] where 𝔽 is any field of characteristic zero, we consider S-modules M = ⊕_{k≧k_{0}}M_{k} which are quasi-finitely generated in the sense that dim_{𝔽} M_{k} < ∞ for all k. The Δ-tensor product M ⊗_{Δ} N of two such modules is defined as a graded 𝔽-vector space by M ⊗_{Δ} N = M ⊗_{𝔽} N and the S-modulde structure is defined by the rule

x^{i}(m⊗_{Δ}n) = x^{i}m⊗_{Δ}n + m⊗_{Δ}x^{i}n.

From the Δ-tensor powers ⊗_{Δ}^{p}M we may define the Δ-exterior powers Λ_{Δ}^{p}M and our main algebraic result is: If M is an involutive S-module with resolution length ℓ(M), then Λ_{Δ}^{p}M is involutive with resolution length ℓ(Λ_{Δ}^{p}M) ≦ ℓ(M). In spite of its “functorial” appearance, this result does not seem to admit of a simple proof, but involves a careful study of the defining relations of the modules in question. That this result is not entirely trivial is suggested by the observation that if the symbol is “split” then the minimal resolution of M essentially lifts to one of the Λ_{Δ}^{p}M, whereas if the symbol is generic then Λ_{Δ}^{p}M is free in the lower degrees and in general is “more free” than M.

The integer ℓ in (1) turns out to be the resolution length of the symbol module M associated to the exterior differential system, and the above vanishing result ultimately follows from the commutative algebra vanishing result that

H_{Kos}^{q}(Λ_{Δ}^{p}M) = 0, p > 0 and 0 ≦ q < n − ℓ

0.1.3 where H_{Kos}^{q} is the Poincaré dual of ordinary Koszul homology.

The quotient group H̄^{n−ℓ} turns out to have a more concrete realization than just as a cohomology group. This comes from the fact that there are canonically defined vector bundles E_{0} and E_{1} (whose fibres are Spencer-type cohomology groups) and a canonical linear differential operator

∇ : E_{0} → E_{1}

such that

(3) H̄^{n−ℓ} ≅ ker ∇.

Thus this cohomology, which is defined as a quotient space, is alternatively given as the kernel of an intrinsic differential operator. In the classical or unconstrained case ℓ = 0, the group H̄^{n} is the space of equivalence classes of functionals on integral manifolds and (3) represents a functional by its Euler-Lagrange equation. The general case is an extension of this concept.

This explicit realization of the space of conservation laws as the kernel of a (generally overdetermined) differential operator will be seen to have important consequence. In particular, one application of (3) is to show that the space of translation-invariant quadratic conservation laws for a determined, linear constant-coefficient PDE system is given by Sym_{Δ}^{2}(M), where M is the module associated to the symbol of the equation.

Finally, in Section 6 we will discuss some topological issues related to (1). The first is a straightforward globalization of (1), expressed by the statements

(i)  H^{q}(X, ℝ) ≅ H̄^{q}(X),  0 ≦ q < n − ℓ
(ii)  0 → H^{n−ℓ}(X, ℝ) → H̄^{n−ℓ}(X) → H^{0}(ℋ̄^{n−ℓ})

where ℋ̄^{n−ℓ} is the sheaf of conservation laws. This result suggests certain purely geometric considerations. If we define H_{∗,ℐ}(X) to be the homology computed from the complex of piecewise-C^{1} chains whose individual simplices are integral manifolds of ℐ, then (i) and (ii) suggest the statements H_{q,ℐ}(X) ≅ H_{q}(X), 0 ≦ q < n − ℓ (iii) H_{n−ℓ,ℐ}(X) → H_{n−ℓ}(X) → 0 . (iv)

0.1.4 Both of these would follow from the local result

H_{q,ℐ}(X) = 0, 0 ≦ q < n − ℓ (v)

where X is a sufficiently small neighborhood of an ordinary integral element. Now (v) is in turn closely related to a result of Thom [10], and in Section 6 we present a sketch of how (v) might follow from Thom’s arguments in case ℐ has no real Cartan characteristics in the range 0 ≦ q < n − ℓ. In general however, the real Cartan characteristics will contribute singularity considerations which Thom’s arguments do not seem to be able to address, thus we only pose (v) as a question.

Especially interesting is the local group H_{n−ℓ,ℐ}(X) and the natural mapping

μ: 𝒞 → Hom(H_{n−ℓ,ℐ}(X), ℝ)

under which conservation laws give what are called moment conditions. We conclude this section by analyzing μ in a number of examples. It is clear that the study of the postulated “ℐ-de-Rham theorem” – meaning the analysis of whether the natural pairing

H̄^{k}(X) ⊗ H_{k,ℐ}(X) → ℝ

is non-degenerate (even locally) – is extremely interesting. The discussion in Section 6 raises more questions than it answers and, in our opinion, points out a very fruitful area for further work.

The principal applications of the general theory will be given in subsequent papers. In particular, in Part II we shall completely analyze the conservation laws for a class of exterior differential systems that we call parabolic systems. As we shall see there the general theory will serve to “guide” the application of E. Cartan’s equivalence method. This method gives in principle an algorithm for determining the intrinsic invariants of an exterior differential system. However, as in classical invariant theory the calculations very quickly get out of hand unless one is studying a situation that is guided by a geometric problem. It is such a guide that is provided by the general theory.

This work has been principally influenced by the papers [13], [14] and [15] of Vinogradov. In seeking to understand the characteristic cohomology of an exterior differential

0.1.5 system we were led to what is now called the “Vinogradov spectral sequence of the variational bicomplex” in the context of an exterior differential system. In this setting there is no longer a bicomplex, but rather there is a filtered complex. By insisting on full contact invariance of the theory one is led naturally to the Frobenius extension construction mentioned above and its cohomological implications.

The papers of Vinogradov built on earlier work of the Russian school of formal differential geometry initiated by Gelfand and his collaborators (see [5] and Manin [8]). There is by now a considerable literature on these subjects, and although our context is quite different we should like to mention Olver [9], Tsujishita [11], Anderson [1] and Dickey [4] as sources for further work and additional references to the literature. In particular, in the case when an exterior differential system arises from a determined PDE system in Cauchy-Kowaleska form, our main result implies the so-called “two-line theorem” of Vinogradov. Extensions of this result, still in the context of the variational bicomplex, to some more general classes of PDE’s have independently been given by Tsujishita [12] and Anderson (loc. cit.).

0.1.6 1. Basic Considerations and First Examples 1.1 Definition of characteristic cohomology

Let X_{0} be a smooth manifold and Ω^{∗}(X_{0}) = ⊕_{p≧0} Ω^{p}(X_{0}) the differential graded algebra of smooth differential forms on X_{0}. We recall that an exterior differential system is given by a homogeneous differential ideal ℐ_{0} ⊂ Ω^{∗}(X_{0}). These two conditions mean that ℐ_{0} = ⊕_{q≧0} ℐ_{0}^{q}, where ℐ_{0}^{q} = ℐ_{0} ∩ Ω^{q}(X_{0}) and dℐ_{0} ⊆ ℐ_{0}. We shall follow the terminology, and for the most part the notations, of [2].

We shall be interested in the integral manifolds of ℐ_{0}. By an integral manifold we shall mean a smooth immersion

f: N → X

where

f^{∗}θ = 0, θ ∈ ℐ_{0}.

Intuitively, integral manifolds are solutions to the PDE system

θ = 0 where θ ∈ ℐ_{0},

and where to “solve the equation θ = 0” means to find a submanifold on which θ restricts to zero. We will in particular be interested in integral manifolds of dimension n, and for purposes of exposition shall make the assumptions

(1)  \{ \begin{matrix}ℐ_{0}^{0} = (0) \\ ℐ_{0}^{q} = Ω^{q}(X_{0}) & for q > n \end{matrix}

The first assumption means that ℐ_{0} contains no functions—otherwise we may replace X_{0} by the subset, assumed to be a submanifold, defined by setting equal to zero all of the functions in ℐ_{0}. Neither assumption changes the set of n-dimensional integral manifolds.

The very definition of an exterior differential system suggests that we consider the complex {Ω̄_{0}^{∗}, d̄} where

\{ \begin{matrix}Ω̄_{0}^{∗} & = Ω^{∗}(X_{0})/ℐ_{0} \\ d̄ & = d mod ℐ_{0} \end{matrix}

Provisional Definition 1.1. The characteristic cohomology H̄_{0}^{∗} of the exterior differential system ℐ_{0} is by definition the cohomology of the complex {Ω̄_{0}^{∗}, d̄}

H̄_{0}^{∗} = H{Ω̄_{0}^{∗}, d̄}.

1.1.1 Elements φ ∈ H̄_{0}^{∗} are called characteristic classes of ℐ_{0}.

We shall explain the subscript “0” in Paragraph 1.3 below. The groups H̄_{0}^{∗} are only a first approximation to our ultimate objects of interest, which will be defined there and denoted by H̄^{∗} without the subscript.

If f : N → X_{0} is an integral manifold of ℐ_{0}, there is an induced mapping

f^{∗}: H̄_{0}^{∗} → H_{DR}^{∗}(N).

We may think of f^{∗}(H̄_{0}^{∗}) as being the cohomology induced on N by virtue of its being a solution to the PDE system. Obvious questions are to “compute” H̄_{0}^{∗}, to understand how the image of f^{∗} varies with f, and so forth.

The two extreme cases H̄_{0}^{n} and H̄_{0}^{0} may be fairly easily interpreted. For example, in case N is compact, using our assumption (1), we have that

H̄_{0}^{n} = Ω^{n}(X_{0})/dΩ^{n−1}(X_{0}) + ℐ_{0}^{n},

and each λ ∈ Ω^{n}(X_{0}) defines a functional I_{λ} on the set of compact integral manifolds by

I_{λ}(f) = ∫_{N} f^{∗}(λ).

Clearly, I_{λ} depends only on the class [λ] ∈ H̄_{0}^{n} defined by λ. The question of how I_{λ} varies with f clearly involves the Euler-Lagrange equations of that functional. Example 1: Pfaffian systems. The understanding of H̄_{0}^{0} is most immediate when ℐ_{0} is a Pfaffian differential ideal, i.e., it is locally generated as a differential ideal in degree one. Concretely, locally there are linear differential forms θ^{1}, …, θ^{s} such that ℐ_{0} is generated algebraically by θ^{1}, …, θ^{s}; dθ^{1}, …, dθ^{s}; we write

ℐ_{0} = {θ^{1}, …, θ^{s}; dθ^{1}, …, dθ^{s}} .

The derived flag ℐ_{0} ⊃ ℐ_{0}^{(1)} ⊃ ℐ_{0}^{(2)} ⊃ ⋯ is by definition the nested sequence of Pfaffian systems defined inductively by ℐ_{0}^{(k+1)} = {θ ∈ ℐ_{0}^{(k)}: dθ ∈ ℐ^{(k)}}. Then ℐ_{0}^{(∞)} = ∩_{k} ℐ_{0}^{(k)} is the

1.1.2 largest completely integrable subsystem of ℐ_{0}. Under suitable constant rank assumptions, we may invoke the Frobenius theorem to find local functions f_{1}, …, f_{r} such that

ℐ_{0}^{(∞)} = {df_{1}, …, df_{r}}.

The ℝ-vector space of functions f satisfying df ∈ ℐ_{0} is classically called the space of first integrals of the exterior differential system, denoted here by F I(ℐ_{0}). Assumption (1) together with the exact cohomology sequence of 0 → ℐ_{0} → Ω^{∗}(X_{0}) → Ω̄^{∗} → 0 gives immediately that locally

H̄_{0}^{0} ≅ {g ∈ Ω^{0}(X_{0}): dg ∈ ℐ_{0}} = F I(ℐ_{0}).

(The assumption that we are working locally is used not only to represent ℐ_{0}^{(∞)} globally in the form {df_{1}, …, df_{r}} but also to insure that H_{DR}^{1}(X_{0}) = 0.) Example 2: Contact Manifolds. A more substantial and interesting example is that of a Pfaffian system locally generated by a single 1-form θ_{0}. With suitable constant rank assumptions, there is an integer n so that

θ_{0} ∧ (dθ_{0})^{n} ≠ 0
θ_{0} ∧ (dθ_{0})^{n+1} = 0.

For simplicity, we are going to assume that dim X_{0} = 2n + 1. In this case, the ideal ℐ_{0} generated by θ_{0} is called a contact ideal and the pair (X_{0}, ℐ_{0}) is called a contact manifold. By the Pfaff-Darboux theorem, every point of X_{0} lies in a neighborhood on which there exist local coordinates (x^{1}, …, x^{n}, z, y_{1}, …, y_{n}) in which a non-zero multiple θ of θ_{0} has the normal form

θ = dz − ∑_{i=1}^{n} y_{i}dx^{i}

Maximal integral manifolds of θ have dimension n and are called Legendre submanifolds of X_{0}. Those on which dx^{1}∧ … ∧dx^{n} ≠ 0 are locally of the form

(x^{1}, …, x^{n}) → (x^{1}, …, x^{n}, z(x), \frac{∂z(x)}{∂x^{1}}, …, \frac{∂z(x)}{∂x^{n}})

1.1.3 We will show that, locally,

(2)  \{ \begin{matrix}H̄_{0}^{0} ≅ ℝ \\ H̄_{0}^{q} = 0, q > 0, q ≠ n \\ dim H̄_{0}^{n} = ∞ \end{matrix}

On contact submanifolds of the form (2), H̄_{0}^{n} may be thought of as equivalence classes of first order functionals

z(x) → ∫ λ (x^{1}, …, x^{n}, z(x), \frac{∂z(x)}{∂x^{1}}, …, \frac{∂z(x)}{∂x^{n}}) dx^{1} ∧ … ∧ dx^{n}.

The proof of (2) will be given following a preliminary linear algebra discussion. Suppose that V is a vector space of dimension 2n + 1 and that we are given

\{ \begin{matrix}θ ∈ V \\ Θ ∈ Λ^{2}V \end{matrix}

satisfying the conditions that (i) Θ is well-defined up to adding multiples α∧θ of θ, and (ii) θ∧Θ^{n} ≠ 0. (We are obviously thinking of V as being a typical T_{p}^{∗}X_{0}.) Set V̄ = V/ℝθ and let Θ̄ ∈ Λ^{2}V̄ be induced by Θ. Then Θ̄ is a non-degenerate 2-form on V̄ and it is well known that there is a so-called “primitive” or “Lefschetz” decomposition of Λ^{∗}V̄. Recall that this comes about as follows: First, one shows that

(i)  Θ̄^{k}: Λ^{n−k}V̄ →^{∼} Λ^{n+k}V̄

is an isomorphism for 1 ≤ k ≤ n. Next, if for k ≧ 0 one defines the primitive space by

P̄^{n−k} = ker {Θ̄^{k+1}: Λ^{n−k}V̄ → Λ^{n+k+2}V̄ } .

Then there is a Lefschetz decomposition

(ii)  Λ^{n−k}V̄ = P̄^{n−k} ⊕ Θ̄·P̄^{n−k−2} ⊕ Θ̄^{2}·P̄^{n−k−4} ⊕ · · · .

If we now let I ⊂ Λ^{∗}V be the homogeneous ideal generated by θ and Θ, it follows from (i) that

(iii)  I^{n+k} = Λ^{n+k}V,  k ≧ 1.

1.1.4 Moreover, from (ii) we see that for k ≧ 0 each ψ ∈ Λ^{n−k}V has an expression

(iv)  ψ = θ ∧ α + β_{0} + Θ ∧ β_{1} + Θ^{2} ∧ β_{2} + ⋯

where the β_{i} are unique modulo θ and satisfy Θ^{k+2i+1}∧β_{i} ≡ 0 mod θ.

We now apply this discussion where V is a typical T_{p}^{∗}X_{0}, the 1-form θ is given by dz − ∑ y_{i} dx^{i}, and Θ = dθ = − ∑ dy_{i}∧dx_{i}. We shall denote by

P^{n−k} ⊂ Ω^{n−k}(X_{0}), k ≧ 0

the space of forms that are primitive modulo θ, i.e., that satisfy

Θ^{k+1} ∧ α ≡ 0 mod θ.

Using (iv), we have a decomposition of all forms in Ω^{n−k}(X_{0}) in terms of multiples of θ and primitive forms.

From (iii) it follows that ℐ_{0}^{n+k} = Ω^{n+k}(X_{0}) for all k ≧ 1 which clearly implies that Ω̄{n+k \atop 0} = 0 for all k ≧ 1 (cf. (1) above). This trivially implies that H̄_{0}^{q} = 0 for q ≧ n + 1.

From the exact cohomology sequence of

0 → ℐ_{0} → Ω^{∗}(X_{0}) → Ω̄_{0}^{∗} → 0,

and remembering that we are working locally (so that H^{q}(X_{0}) = 0 for q > 0), we see that the coboundary map induces isomorphisms

(3)  δ: H̄_{0}^{q} →^{∼} H^{q+1}(ℐ_{0}),  q ≧ 1.

Using this we will first prove that H̄_{0}^{q} = 0 for 0 < q < n by showing that

(4)  H^{p}(ℐ_{0}) = 0,  0 ≦ p ≦ n.

Let

ψ = θ ∧ α + Θ ∧ β ∈ ℐ_{0}^{p}

1.1.5 be closed. Then α, dβ ∈ Ω^{p−1}(X_{0}) and

0 = Θ ∧ (α + dβ) − θ ∧ dα
⟹ α + dβ = θ ∧ γ  by (i)
⟹ ψ = d(θ ∧ β)

which gives (4).

We now turn to the most interesting group H̄_{0}^{n}. We will show that a closed form η ∈ ℐ_{0}^{n+1} (= Ω^{n+1}(X_{0})) has an expression

(5)  \{ \begin{matrix}η ≡ θ∧π mod dℐ_{0}^{n} \\ π ∈ P^{n} \end{matrix}

where π is unique mod θ and depends only on the class [η] ∈ H^{n+1}(ℐ_{0}) of η. Moreover, given π ∈ P^{n} we have by definition that

Θ ∧ π = θ ∧ γ.

Replacing π by π +θ∧α gives γ → γ +Θ∧α. Thus, we may uniquely determine π by further requiring that Θ∧π = 0. The condition that η = θ∧π be closed is then that

(6)  dπ ∧ θ = 0

Thus

H̄_{0}^{n} ≅ H^{n+1}(ℐ_{0}) ≅ {π ∈ P^{n} mod θ: dπ ∧ θ = 0}

is naturally represented as the sections of a vector bundle (actually a sub-bundle of the differential forms on X_{0}) satisfying a linear differential equation. (In the involutive case, this sort of “subspace” rather than “quotient” representation of H̄_{0}^{n} will be a characteristic feature of the first non-vanishing H̄^{q} with q > 0, once these groups have been defined.)

Turning to the proof of (5), write

η = θ ∧ α + Θ ∧ β

Then, as before, dΘ = 0 gives that

Θ ∧ (α + dβ) ≡ 0 mod θ
⟹ α + dβ = π + θ ∧ γ

1.1.6 where π ∈ P^{n} is primitive mod θ

(7)  ⟹ η = d(θ ∧ β) + θ ∧ π.

To show uniqueness we suppose that we have

dξ = θ ∧ π

for some ξ = θ∧ρ + Θ∧σ ∈ ℐ^{n}. This gives

Θ ∧ (ρ + dσ) − θ ∧ dρ = θ ∧ π
⟹ ρ + dσ = θ ∧ γ  for some γ
⟹ ξ = d(θ ∧ σ)
⟹ θ ∧ π = 0.

Thus, π is unique mod θ and depends only on the class [η] ∈ H^{n+1}(ℐ_{0}), as we wished to show.

Using the isomorphism (3) we have

δ: H̄_{0}^{n} →^{∼} {π ∈ P^{n} mod θ: (6) is satisfied}

As we shall now explain, the mapping δ is a second order linear differential operator which may be interpreted as the Euler-Lagrange equations associated to the functional defined by ψ̄ ∈ Ω̄_{0}^{n}.

Given ψ̄ we seek a lifting ψ ∈ Ω^{n}(X_{0}) such that

(8)  dψ = θ ∧ π

where π ∈ P^{n}. If ψ′ is any lifting of ψ̄, then as in (7) we have

dψ′ = θ ∧ π + d(θ ∧ β)
⟹ d(ψ′ − θ ∧ β) = θ ∧ π

Taking ψ = ψ′ − θ∧β gives (8). We note that each of the mappings

\{ \begin{matrix}ψ̄ → ψ \\ ψ → π \end{matrix}

1.1.7 are 1^{st}-order linear differential operators, so that ψ̄ → π is second order as claimed.

In Section 3.3 we will further discuss the infinite prolongation of this example, in effect giving a resolution of H̄^{n} by a complex of canonical linear, first order differential operators. Example 3: Symplectic manifolds. Now let us suppose that dim X_{0} = 2n and that Θ is a closed 2-form on X_{0} which satisfies the condition that Θ^{n} is nowhere vanishing, i.e., that Θ defines a symplectic structure on X_{0}. We shall consider the differential ideal ℐ_{0} generated by the non-degenerate 2-form Θ. Also, for simplicity, we shall exclude the trivial case n = 1 and henceforth assume that n > 1. By the Pfaff-Darboux theorem, in suitable local coordinates (x^{1}, …, x^{n}, ξ_{1}, …, ξ_{n}) on X_{0} we have

Θ = dθ = ∑_{i=1}^{n} dξ_{i} ∧ dx^{i}.

The maximal integral manifolds of Θ have dimension n and are generally referred to as Lagrangian submanifolds of X_{0}. Those on which dx^{1}∧ … ∧dx^{n} ≠ 0 are locally of the form

(x^{1}, …, x^{n}) → (x^{1}, …, x^{n}, \frac{∂S(x)}{∂x^{1}}, …, \frac{∂S(x)}{∂x^{n}})

where S(x) is an arbitrary smooth function. We will show that, locally,

H̄_{0}^{q} = 0 for q ≠ 0, 1, or n
H̄_{0}^{0} ≅ ℝ
H̄_{0}^{1} ≅ ℝ
dim H̄_{0}^{n} = ∞.

(The case n = 1 is somewhat analogous but requires a separate discussion). The proof is very similar to the contact manifold case and will only be sketched.

From Θ^{k}: Ω^{n−k}(X_{0}) →^{∼} Ω^{n+k}(X_{0}), k ≧ 1, we have ℐ_{0}^{n+k} = Ω^{n+k}(X_{0}) for k ≧ 1 and as a trivial consequence H̄_{0}^{q} = 0 for q > n.

As in (3) we have isomorphisms

δ: H̄_{0}^{q}: →^{∼} H^{q+1}(ℐ_{0})

1.1.8 for q > 0. If

ψ = Θ ∧ α ∈ ℐ_{0}^{p}

is closed, then

0 = dψ = Θ ∧ dα.

Since dα ∈ Ω^{p−1}(X_{0}) we infer that dα = 0 if p ≦ n. Thus, if 3 ≦ p ≦ n, then we may write locally α = dβ, which gives ψ = d(Θ∧β), which implies that 0 = H^{p}(ℐ_{0}) ≅ H̄_{0}^{p−1}. If p = 2 then α = C is a constant and ψ = CΘ, from which we infer that H̄_{0}^{1} ≅ ℝ. Finally, if p = n + 1 then dα = π is a closed, primitive n-form. It is easily checked that π depends only on the class [ψ] ∈ H^{n+1}(ℐ_{0}). This gives the identification

δ: H̄_{0}^{n} →^{∼} {closed, primitive n-forms} .

Again, H̄_{0}^{n} is represented as the kernel of a canonical linear differential operator, modulo nothing.

Remark: If locally we choose a one form α with dα = Θ, then the most general solution of dθ = Θ is θ = α+df for an arbitrary function f. This suggests that we consider the “universal” solution to this problem, which is obtained by taking

Y = X_{0} × ℝ

where ℝ has coordiante u and setting

θ = du − α

on Y. If 𝒥 ⊂ Ω^{∗}(Y) is the exterior differential system generated by θ, then the projection π: Y → X_{0} induces mappings

Ω^{∗}(X_{0}) ↪ Ω^{∗}(Y)
∪  ∪
ℐ_{0} ↪ 𝒥

which give

π^{∗}: H̄^{q}(X_{0}) ⟶ H̄_{0}^{q}(Y).

this mapping π^{∗} is an isomorphism for q ≠ 0, 1, or n, while π^{∗} kills H̄_{0}^{1}(X_{0}).

1.1.9

This process of adjoining a “primitive” or potential (think of u as ∫ α on integral manifolds of θ = 0) will reappear and be formalized in Part II in connection with conservation laws of certain specific parabolic systems (such as Burger’s equation).

We will conclude this section with a discussion of two related examples. Example 4: Complex manifolds. Let X_{0} be a complex manifold, with Ω^{p,q}(X_{0}) denoting the space of smooth (p, q)-forms on X_{0}. We let ℐ_{0} be the differential ideal generated by the 2-forms

θ = α + ᾱ, α ∈ Ω^{2,0}(X_{0}).

Clearly, ℐ_{0}^{1} = 0 and ℐ_{0}^{q} = Ω^{q}(X_{0}) for q≥3. Integral manifolds of ℐ_{0} are thus of (real) dimension at most two, and those of dimension two are holomorphic curves in X_{0}. The first interesting characteristic cohomology group is H̄_{0}^{1}, and denoting by Ω_{c}^{p,0}(X_{0}) the closed (and therefore holomorphic) (p, 0) forms, we shall show that

(9)  H̄_{0}^{1} ≅ Ω_{c}^{2,0}(X_{0}).

Proof: A class in H̄_{0}^{1} is represented by

η = η′ + η̄′, η′ ∈ Ω^{1,0}(X_{0})

which satisfies

¯∂η′ + ∂η̄′ = 0.

Then

π = ∂η′ ∈ Ω^{2,0}(X_{0})

satisfies

¯∂π = −∂ ¯∂η′ = ∂^{2}η̄′ = 0.

It is easily seen that π depends only on the class of η ∈ H̄_{0}^{1}, and that any π ∈ Ω_{c}^{2,0}(X_{0}) arises from a class in H̄_{0}^{1}. Q.E.D.

Again we encounter the phenomenon that the first non-vanishing characteristic cohomology group is isomorphic to the kernel of a canonical linear differential operator, modulo nothing.

1.1.10

As will be made clear when we discuss conservation laws, in this example, the group H̄_{0}^{1} can be interpreted as the space of independent conservation laws for this system. Note that when X_{0} = ℂ^{n}, this space is of infinite dimension. This is perhaps the simplest non-trivial example of an exterior differential system with an infinite number of independent local conservation laws. Ultimately these are rooted in the Cauchy integral formulas

\frac{d}{dt}(\frac{1}{2πi} ∫_{|z|=t} z^{k}f(z)dz) = 0

for holomorphic functions in the complex plane. Example 5: Associative manifolds. Our last example is based on Bryant [3] and especially on work of Harvey-Lawson work [7] on calibrated geometries. We denote by 𝕆 the octonions with standard basis e_{0}, e_{1}, …, e_{8} where e_{0} is the unit. We then let ℝ^{7} = Im 𝕆 be the imaginary octarians with the standard metric having e_{1}, …, e_{7} as an orthonormal basis. Using x^{1}, …, x^{7} as dual linear coordiantes on ℝ^{7} and setting

dx^{ijk} = dx^{i} ∧ dx^{j} ∧ dx^{k},

we consider the 3-form

φ = dx^{123} + dx^{145} + dx^{167} + dx^{246} − dx^{257} − dx^{347} − dx^{356}

which gives the multiplication table of Im 𝕆 in the following manner:

(i) e_{i}^{2} = −e_{0}, (ii) for i, j distinct

e_{i} · e_{j} + e_{j} · e_{i} = 0,

(iii) for i, j k distinct we have

e_{i} · e_{j} = e_{k}

if, and only if, dx^{ijk} appears with a + sign in φ.

The form φ gives a calibration in the sense that for any vectors v_{1}, v_{2}, v_{3}

(10)  |φ(v_{1}, v_{2}, v_{3})| ≦ |v_{1} ∧ v_{2} ∧ v_{3}|

1.1.11 with equality holding if, and only if, v_{1}, v_{2}, v_{3} span a 3-plane that is closed under multiplication (these are called associative 3-planes). We note the analogy to Kähler geometry, where for

ψ = \frac{\sqrt{−1}}{2} ∑_{i=1}^{n} dz^{i} ∧ dz̄^{i} = ∑_{i=1}^{n} dx^{i} ∧ dy^{i}

in ℂ^{n} with coordinates z^{i} = x^{i} + \sqrt{−1} y^{i}, we have Wirtinger’s inequality

|ψ(v_{1}, v_{2})| ≦ |v_{1} ∧ v_{2}|

with equality if and only if v_{1} and v_{2} are either linearly dependent or span a complex line. By further analogy, we will define an exterior differential system on ℝ^{7} whose 3-dimensional integral elements are exactly the associative 3-planes.

For this we set

Θ_{i} = e_{i} ∗φ

and recall from the references given above that (10) may be completed into an equality by virtue of the relation

|φ(v_{1}, v_{2}, v_{3})|^{2} + ∑_{i=1}^{7} |Θ_{i}(v_{1}, v_{2}, v_{3})|^{2} = |v_{1} ∧ v_{2} ∧ v_{3}|^{2}.

Thus the exterior equations

Θ_{i} = 0

generate an exterior differential system ℐ_{0} on ℝ^{7} whose 3-dimensional integral elements are exactly the associative 3-planes. We note that

ℐ_{0}^{q} = 0 q = 0, 1, 2,

and it may be shown that

ℐ_{0}^{q} = Ω^{q}(ℝ^{7}) q ≧ 4.

Thus the interesting characteristic cohomology group is H̄_{0}^{2}, and we will show that

(11)  dim H̄_{0}^{2} = 21

1.1.12 Proof: It is clear that

H̄_{0}^{2} ≅ closed forms in ℐ_{0}^{3}.

Now any form in ℐ_{0}^{3} is

ψ = X ∗φ

for some vector field X, and from

0 = dψ = d(X ∗φ) = ℒ_{X}(∗φ),

we see that exp(tX) preserves the form ∗φ. We will show that

(i) the mapping φ → ∗φ is 2 : 1, so that exp(tX) preserves φ; and (ii) φ determines the metric, so that X is an infinitesimal Euclidean motion.

This proves already that dim H̄_{0}^{2} < ∞, and the fact that dim H̄_{0}^{2} = 21 will depend on knowing that the symmetry group of φ is the exceptional group G_{2}, which has dimension 14.

Let V be a 7-dimensional real vector space and φ ∈ Λ^{3}V^{∗}. We note that this is one of the very few cases (other than the obvious ones) where GL(V) acting on Λ^{q}V^{∗} has an open orbit. For φ as above we define a symmetric bilinear map

b_{φ}: V × V ⟶ Λ^{7}V^{∗}

by

b_{φ}(v, w) = \frac{1}{6}(v φ) ∧ (w φ) ∧ φ.

We say that φ is non-degenerate if b_{φ} is non-singular, and observe that the φ given above has this property. To convert b_{φ} into a metric we need to single out a canonical volume form, which we now do by singling out unimodular bases for V. If v_{1}, …, v_{7} is any basis we set

B_{φ}^{v}(v_{i}, v_{j}) = b_{φ}(v_{i}, v_{j})(v_{1}, …, v_{7}).

If A ∈ GL_{7}(ℝ) transforms v_{i} to ṽ_{i} then a calcualtion shows that

det_{i,j} ∥Bṽ ; φ(v_{i}, v_{j})∥ = (det A)^{9} det_{i,j} ∥B_{φ}^{v}(v_{i}, v_{j})∥

1.1.13 For non-degenerate φ, we will say that a basis v_{1}, …, v_{7} of V is φ-unimodular in case det_{i,j} ∥B_{φ}^{v}(v_{i}, v_{j})∥ = 1. In this way a non-degenerate φ gives a metric on ℝ^{7}. For the particular φ given above,

B_{φ}^{e}(e_{i}, e_{j}) = δ_{ij}

is the standard metric.

The fact that the mapping φ → ∗φ is one-to-one near our particular φ (which is all that we need) is easily shown by computing the differential of the ∗ mapping at φ. It follows that any 1-parameter subgroup of diffeomorphisms of ℝ^{7} which preserves ∗φ must also preserve ϕ and the standard flat metric. Since the group of diffeomorphisms which preserve φ clearly contains the translations, it follows that this group must be the semidirect product of the translations with the subgroup of those rotations about the origin which preserve φ.

At this stage we have shown that

X = T + L

where T is a translation vector field (corresponding to constant linear combinations of the Θ_{i}) and L ∈ 𝔤𝔩(ℝ^{7}) preserves φ. But since φ determines the multiplication in Im 𝕆 it follows that L induces an infinitesimal automorphism of 𝕆, and it is then known that L must lie in 𝔤_{2} ⊂ 𝔤𝔩(ℝ^{7}) where 𝔤_{2} is the Lie algebra of G_{2} embedded in 𝔤𝔩(ℝ^{7}) by the standard representation G_{2} → GL(Im 𝕆).

Remark: We have shown that for any connected open set U ⊂ ℝ^{7} the characteristic cohomology H̄_{0}^{2}(U) is a 21-dimensional vector space. In a sense this example therefore resembles more the symplectic example above than the complex manifold example. However, in Section 6 below we will see that the differences are more important than the similarities.

1.1.14 1.2 The exterior differential system associated to a partial differential

equation; symmetries

It is well known that any partial differential equation system – PDE system for short – may be canonically rewritten as an exterior differential system in such a way that the solutions to the PDE system give integral manifolds of the exterior differential system. The converse is also true provided that we work locally and impose a transversality condition on integral manifolds. However, the exterior differential system formulation gives a coordinate free method of studying the system; moreover, the larger group of symmetries of the exterior differential system will allow us to do calculations of examples not yet possible in the coordinate formulation. In this section we will briefly review this construction. For simplicity of exposition we will restrict ourselves to the case of a first order PDE system, remarking that similar considerations apply to the general situation.

The most classical formulation of a PDE system is to give a submanifold R, satisfying suitable conditions to be specified below, of the manifold J^{1}(N, Y) of 1-jets of smooth mappings from a manifold N to a manifold Y. In this formulation, the symmetries are the diffeomorphisms of R that preserve the two coordinate projections R → N and R → Y. Concretely, let x^{1}, …, x^{n} and u^{1}, …, u^{s} be local coordinates on N and Y, respectively. Then, using the ranges of indices

1 ≦ i, j ≦ n, 1 ≦ α, β ≦ s,

there are induced local coordinates {x^{i}, u^{α}, p_{i}^{α}} on J^{1}(N, Y). Symmetries are induced by diffeomorphic changes of independent and dependent variables

(1)  \{ \begin{matrix}x′^{i} & = x′^{i}(x) \\ u′^{α} & = u′^{α}(u) \\ p′_{i}^{α} & = ∂u′^{α}/∂u^{β}p_{j}^{β}∂x^{j}/∂x′^{i} \end{matrix}

Each mapping f: N → Y given locally by x^{i} → u^{α}(x) has a canonical lifting j_{1}(f) ⊂ J^{1}(N, Y) given in coordinates by {x^{i}, u^{α}(x), ∂u^{α}/∂x^{i}}. That is, J^{1}(N, Y) is just the vector bundle Hom(T N, T Y) lying over N ×Y, and associated to a mapping f: N → Y is first of all the graph Γ_{f} ⊂ N × Y of f, and then lying over this is the graph j_{1}(f) ⊂ J^{1}(N, Y) of the differential of f.

1.2.1

A submanifold R ⊂ J^{1}(N, Y) may be thought of as imposing constraints on the differentials of maps from N to Y; i.e., as a PDE system. In coordinates, R is given locally by equations

(2)  F^{λ}(x^{i}, u^{α}, p_{i}^{α}) = 0,

to which corresponds the PDE system

(3)  F^{λ}(x^{i}, u^{α}(x), ∂u^{α}(x)/∂x^{i}) = 0.

We assume that the equations (2) define a submanifold R ⊂ J^{1}(N, Y) and that the projection R → N is a submersion. As explained above, the group Diff N × Diff Y acts as a group of diffeomorphisms of J^{1}(N, Y), and the subgroup preserving R is by definition the classical symmetry group of the PDE system (3).

A generalization arises by considering a fibered manifold Z → N and defining a PDE system to be given by a submanifold R ⊂ J^{1}(N, Z) where J^{1}(N, Z) is the manifold of 1-jets of smooth sections of the fibered manifold. By definition of a fibered manifold, the projection Z → N is a submersion, and locally on Z there are product coordinates (x^{i}, u^{α}) such that Z → N is given by (x^{i}, u^{α}) → (x^{i}). Then there are induced local coordinates (x^{i}, u^{α}, p_{i}^{α}) on J^{1}(N, Z), and R given by (2) corresponds to the PDE system (3).

The difference between these two constructions is that in the fibered manifold context the symmetry group is enlarged to include gauge transformations. That is, the automorphism group of J^{1}(N, Z) is induced from the automorphisms of the fibered manifold, defined to be the fiber preserving diffeomorphisms of Z given in local coordinates by

(4)  \{ \begin{matrix}x′^{i} & = x′^{i}(x) \\ u′^{α} & = u′^{α}(u, x) \\ p′_{i}^{α} & = ((∂u′^{α}/∂u^{β})p_{j}^{β} + ∂u′^{α}/∂x^{j})∂x^{j}/∂x′^{α}. \end{matrix}

The enlarged symmetry group of R now consists of all transformations (4) which preserve R.

Associated canonically to either a classical PDE system or to a PDE system imposed on sections of a fibered manifold is an exterior differential system. The underlying manifold

1.2.2 is R, and the exterior differential system is the restriction to R of the contact system defined on either of J^{1}(N, Y) or J^{1}(N, Z). The contact system is the Pfaffian system locally generated by the 1-forms

θ^{α} = du^{α} − p_{i}^{α}dx^{i}.

Of course, the contact system may be described in a coordinate-free manner which makes apparent the invariance under the corresponding symmetry group, but we shall not do this here. An integral manifold of this system on which the transversality condition dx^{1}∧ … ∧dx^{n} ≠ 0 is satisfied is locally a graph

(x^{i}) → (x^{i}, u^{α}(x), p_{i}^{α}(x))

and the vanishing of the contact forms when restricted to this submanifold implies that

p_{i}^{α}(x) = ∂u^{α}(x)/∂x^{i}.

Restricting to R therefore gives an exterior differential system ℐ whose integral manifolds satisfying a transversality condition are locally in one-to-one correspondence with solutions to the PDE system.

However, by definition the symmetries of ℐ are the diffeomorphisms of R that preserve ℐ. The group Aut(ℐ) of these symmetries certainly contains the gauge group described locally by the transformations (4) preserving R, but Aut(ℐ) may be strictly bigger than this gauge group. The constructions in which we will be interested in this paper are all invariant under this larger group.

Moreover, regarding two PDE R_{1} and R_{2} as the same if there is a diffeomorphism R_{1} → R_{2} which identifies the ideals ℐ_{1} and ℐ_{2} is a natural notion of equivalence which properly generalizes gauge equivalence. For example as we shall eventually see, the following equations, though not gauge equivalent to linear equations, lead to manifolds R with ideals ℐ which are (globally) equivalent to those generated by linear PDE:

u_{t} = \frac{au_{xx} + b}{cu_{xx} + d}, ad − bc = −1

u_{xx}u_{yy} − u_{xy}^{2} = −1

u_{xx} − k^{2}(u_{y})u_{yy} = 0, k(u_{y}) ≠ 0

1.2.3 and, more generally,

u_{t} + f(u)u_{x} = 0

where u ∈ ℝ^{2} and the matrix f(u) has everywhere distinct real eigenvalues.

Before going on we would like to mention one piece of terminology. Namely, when R is all of J^{1}(N, Y) or J^{1}(N, Z)—i.e., when there are no equations (2)—we shall say that we are in the unconstrained case. In this case it is a classical result due to Lie and Bäcklund that all symmetries preserving the contact system are in fact gauge transformations (4) when s > 1. When s = 1 the transformations preserving ℐ are only required to preserve the single contact form θ = du − p_{i} dx^{i} up to a scalar factor, and it is well known that this group is strictly larger than (4). Such symmetries may not preserve the above transversality condition and are analogous to the contact transformations of classical mechanics.

We remarked above that Aut(ℐ) may be strictly larger than the gauge group. A discussion to the effect that this is the exception rather than the rule is given in Vinogradov [14]. Vinogradov’s claim is analogous to the statement that an isometry of a generic submanifold of Euclidean space is induced by a Euclidean motion. For our eventual purposes we are interested in non-generic exterior differential systems; e. g., those having a large number of conservation laws. For these the distinction may be more significant. More importantly, we shall use the method of frame adaptation (equivalence method of E. Cartan) to study in practice the conditions imposed an a particular class of exterior differential systems by the requirement that there be conservation laws. This method consists in successively normalizing the invariants of the exterior differential system and refining the frame adaptations—i.e., reducing the structure group—accordingly. These frame adaptations will be made without reference to a particular set of independent variables and are therefore reflective of Aut(ℐ) rather than the smaller group of gauge transformations. The freedom to make such frame adaptations will be absolutely essential for the application of the general theory.

1.2.4 1.3 The prolongation tower

Given an exterior differential system ℐ_{0} on a manifold we have introduced the characteristic cohomology groups H̄_{0}^{q} and remarked that, for example, H̄_{0}^{n} may be interpreted as functionals on integral manifolds. Natural questions that may arise are: “What are the Euler-Lagrange equations of such a functional?”, and the inverse problem “When is a given PDE system the Euler-Lagrange equations of a functional?” It is well known that, in general, the Euler-Lagrange equations of a functional ∫ λ(x, u, ∂u, …, ∂^{k}u)dx will have order 2k, but in a number of very interesting cases (e.g., the Einstein equations) the order may be less. Thus, in the inverse problem we should not specify the order of derivative in the unknown functional, i.e., we should work on the manifold of jets of arbitrary order.

Somewhat more substantively, we shall see that for the exterior differential system arising from a determined PDE system, the group H̄_{0}^{n−1} is essentially the space of conservation laws for the PDE system. As famously illustrated by the KDV equation, these conservation laws may occur at arbitrarily high jet levels. These considerations suggest that we extend the original problem by allowing jets of arbitrarily high order, a process known as prolongation. Thus, given ℐ_{0} on X_{0} under certain regularity assumptions we will pass to the infinite prolongation ℐ on an infinite dimensional manifold X, thereby encapsulating behavior of arbitrarily high order.

More formally, let G_{n}(T X_{0}) → X_{0} denote the bundle whose fiber over a point p ∈ X_{0} is the Grassmannian G_{n}(T_{p}X_{0}) of n-planes in the tangent space T_{p}X_{0}. Over G_{n}(T_{p}X_{0}) there is a canonical exterior differential system denoted by ℒ_{1} whose integral manifolds are just the canonical lifts (Gauss mappings) of immersions f: N → X_{0}:

(1)  \begin{aligned}& & G_{n}(T X_{0}) \\ & ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}f_{∗} & ↓^{π} \\ N & ⟶^{f} & X \end{aligned}

Given E_{0} ∈ G_{n}(T X_{0}) we may choose local coordinates x^{1}, …, x^{n}, u^{1}, …, u^{s} on X_{0} such that E_{0} is given by du^{α} = 0. A neighborhood of E_{0} in G_{n}(T X_{0}) then has local coordinates (x^{i}, u^{α}, p_{i}^{α}) where the corresponding n-plane is given by

θ^{α} = du^{α} − p_{i}^{α}dx^{i} = 0

1.3.1 More intrinsically, for an n-plane E ∈ G_{n}(T_{p}X_{0}) the subspace

π^{∗}(E^{⊥}) ⊂ π^{∗}(T_{p}^{∗}X_{0}) ⊂ T_{E}^{∗}(G_{n}(T X_{0}))

has as basis the 1-forms θ^{α}, and these forms generate the differential ideal giving the canonical exterior differential system ℒ_{1} on G_{n}(T X_{0}). The property summarized by the diagram (1) is apparent.

Next, we recall that an integral element of ℐ_{0} is given by a linear subspace E ⊂ T_{p}X_{0} such that

(2)  θ|_{E} = 0, θ ∈ ℐ_{0}

The set of all n-dimensional integral elements will be denoted by G_{n}(ℐ_{0}) ⊂ G_{n}(T X_{0}). We assume that G_{n}(ℐ_{0}) is a smooth manifold with defining equations (2), and then the first prolongation (X_{1}, ℐ_{1}) of (X_{0}, ℐ_{0}) is defined to be the restriction to G_{n}(ℐ_{0}) of the canonical Pfaffian system ℒ_{1} on G_{n}(T X_{0}). By what has been said above, the integral manifolds of ℐ_{1} and ℐ_{0} are in one-to-one correspondence. In local coordinates, the equations (2) are

F^{λ}(x^{i}, u^{α}, p_{i}^{α}) = 0

where F^{λ} is a function of the special form

∑ f_{α_{1}...α_{k}}^{i_{1}...i_{k}}(x, u) det ∥p_{i_{v}}^{α_{v}}∥,

expressing the important fact that the defining equations of G_{n}(ℐ_{0}) are the exterior equations (2). Thus the case of a classical PDE system, of the PDE system imposed on sections of a fibered manifold, and of the first prolongation of an exterior differential system all “look the same” in appropriate local coordinates, except that in the last case the defining equations are of a special form. In each of the first two cases there is a canonically associated exterior differential system, and all three of the Pfaffian differential systems are generated by 1-forms having the same expression in local coordinates. Although we shall use the PDE system imposed on sections of a fibered manifold for illustrative purposes, it is the last of the above contexts in which we shall be working.

1.3.2

Returning to our general discussion, each higher prolongation (X_{k}, ℐ_{k}) is defined inductively to be the first prolongation of (X_{k−1}, ℐ_{k−1}). This leads to the prolongation tower

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(3)

with the property that the integral manifolds f_{k}: N → X_{k} of ℐ_{k} are in one-to-one correspondence with the integral manifolds f_{0}: N → X_{0} of ℐ_{0}. Intuitively, for q ∈ N the point f_{k}(q) in X_{k} is the k-jet of f_{0} at the point q.

Now we shall make the following Regularity Assumption: The X_{k} are smooth manifolds and X_{k} → X_{k−1} is a surjective submersion. This assumption is satisfied if, e.g., we restrict attention to the projection to X_{0} of a neighborhood of a regular integral element E_{0} (cf. Chapter III of [2]) and take X_{1} to be this neighborhood. That E_{0} should have this property is in practice usually easy to verify using Cartan’s test.

With this assumption there are inclusions

π_{k}^{∗}: Ω^{∗}(X_{k−1}) ⟶ Ω^{∗}(X_{k})
∪  ∪
π_{k}^{∗}: ℐ_{k−1} ⟶ ℐ_{k}

1.3.3 and, omitting reference to the π_{k}^{∗}, we define

X = \underleftarrow{lim} X_{k}
Ω^{∗}(X) = ⋃_{k≧0} Ω^{∗}(X_{k})
ℐ = ⋃_{k≧0} ℐ_{k}

A point of X is an infinite jet of integral manifold of ℐ_{0}, and by definition a differential form on X is a form on some X_{k_{0}} (or, equivalently, forms φ_{k} on X_{k} for k ≧ k_{0} with the coherence property that π_{k}^{∗}φ_{k−1} = φ_{k} for k ≧ k_{0} + 1). The space X is the setting for the “formal differential geometry” of the Russian school and others. The infinite dimensionality of X will not be an issue for us, and we refer to [1] and [11] for general discussions about the rules for doing “calculus” on X.

Our basic object of study is given by the following

Definition: Let ℐ_{0} be an exterior differential system on a manifold X_{0}, and construct the prolongation tower (3). Then the characteristic cohomology H̄^{∗} is by definition the cohomology of the complex {Ω^{∗}(X)/ℐ, d}. Setting Ω̄^{∗} = Ω^{∗}(X)/ℐ and d̄ = d mod ℐ we thus have

H̄^{∗} = H{Ω̄^{∗}, d̄}.

It is useful to examine the prolongation tower construction in local coordinates. We first do this in the unconstrained case – i.e., where ℐ_{0} is zero – and for this we denote by G_{n,k}(X_{0}) the space of k-jets of n-dimensional submanifolds of X. Observe that G_{n,1}(X_{0}) = G_{n}(T X_{0}) in our previous notation, that X_{k} ⊂ G_{n,k}(X_{0}), and that there is a “universal prolongation tower”

1.3.4

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containing (3) as a sub-tower. Generalizing the above local coordinate system on G_{n,1}(X_{0})

– which amounts to locally in G_{n,1}(X_{0}) representing n-dimensional submanifolds of X_{0} as graphs – there are local coordinates

(x^{i}, u^{α}, p_{i}^{α}, p_{ij}^{α}, …, p_{I}^{α}), |I| ≦ k

in G_{n,k}(X_{0}). Here p_{ij}^{α} = p_{ji}^{α}, and in general we are using standard multi-index notation I = (i_{1}, …, i_{k}) where p_{I}^{α} is symmetric in the i_{v} (think of p_{I}^{α} as representing ∂^{k}u^{α}/∂x^{I}). The canonical contact system ℒ_{k} on G_{n,k}(X_{0}) is the Pfaffian differential system generated locally by the 1-forms

θ^{α} = du^{α} − p_{i}^{α}dx^{i}
θ_{i}^{α} = dp_{i}^{α} − p_{ij}^{α}dx^{j}
⋮
θ_{I}^{α} = dp_{I}^{α} − p_{Ii}^{α}dx^{i},  |I| = k − 1

As in the case k = 1 we can give a coordinate free description of ℒ_{k}, which amounts to saying that the θ_{I}^{α}, |I| ≦ k −1 span the annihilator to the span in T G_{n,k}(X_{0}) of all tangent

1.3.5 spaces to k-jets of n-dimensional submanifolds of X_{0}. From

dθ_{J}^{α} = −dp_{Ji}^{α} ∧ dx^{i}
= −θ_{Ji}^{α} ∧ dx^{i}, |J| ≦ k − 2

we obtain the following simple but basic fact: Let Θ_{k} denote the 1 forms in ℒ_{k} and {Θ_{k}} ⊂ Ω^{∗}(G_{n,k}(X_{0})) the algebraic ideal they generate. Then

(4)  dΘ_{k−1} ⊆ {Θ_{k}}.

In other words: The universal prolongation tower is constructed by successively adjoining higher derivatives as new variables. This is reflected in the sequence of Pfaffian systems ℒ_{1} ⊂ ℒ_{2} ⊂ ⋯ by the statement that ℒ_{k−1} satisfies the Frobenius integrability condition (4) relative to ℒ_{k}. In particular, if we pass to jets of infinite order and set

G_{n}(X_{0}) = \underleftarrow{lim} G_{n,k}(X_{0})
ℒ = ⋃_{k≧0} ℒ_{k},

then ℒ is formally a Frobenius system on G_{n}(X_{0}). Since G_{n}(X_{0}) is not a finite dimensional manifold, we cannot of course apply the usual Frobenius theorem to conclude that ℒ defines a true foliation. However, at least informally we may think of the leaves of this “foliation” as being the infinite jets of n-dimensional submanifolds of X_{0}.

By construction, these considerations restrict to the prolongation tower (3) defined by the original exterior differential system ℐ_{0} on X_{0}. With the notations

ℱ = functions on X
Ω^{∗} = Ω^{∗}(X)
Ω_{k}^{∗} = differential graded subalgebra of Ω^{∗} given by Ω^{∗}(X_{k})
Θ_{k} = ℐ_{k} ∩ Ω^{1} = ℐ ∩ Ω_{k}^{1}
{Θ_{k}} = algebraic ideal in Ω^{∗} generated by Θ_{k}

1.3.6 the basic observation is that

dΘ_{k} ⊂ {Θ_{k+1}}.

Since ℐ_{k} is generated as a differential ideal by Θ_{k}, we may think of the construction of the prolongation tower (3) as being reflected algebraically by extensions

ℐ_{k} ⊂ Ω_{k}
∪  ∪
ℐ_{k−1} ⊂ Ω_{k−1}

with the property that ℐ_{k−1} is closed up in Ω_{k} relative to the Frobenius integrability condition. We shall formalize this in Section 2.3 below.

1.3.7 1.4 Variation of characteristic cohomology

Let N be a compact manifold and consider integral manifolds

f: N → X

of ℐ. We shall identify two integral manifolds when the corresponding immersions differ by a diffeomorphism of N, and we denote by ℳ the set of such equivalence classes. We are interested in how the characteristic cohomology f^{∗}(H̄^{∗}) ⊂ H_{DR}^{∗}(N) varies with f, and we observe that this is really a question concerning equivalence classes of integral manifolds. In discussing this question we shall argue formally. Thus, we assume that ℳ is a “reasonable” space and that there is a diagram

\begin{matrix}𝒩 & ⟶^{F} & X \\ π & ↓ \\ ℳ \end{matrix}

where, if for t ∈ ℳ we set

N_{t} = π^{−1}(t)
f_{t} = F |N_{t},

then N_{t} is diffeomorphic to N and under this diffeomorphism f_{t} represents the equivalence class of the integral manifold given by t ∈ ℳ.

We denote by 𝒥 ⊂ Ω^{∗}(𝒩) the differential ideal generated by Ω^{1}(ℳ); the maximal integral manifolds of 𝒥 are just the N_{t}’s. By construction we have

F^{∗}(ℐ) ⊆ 𝒥.

Now for any differential graded algebra {Φ^{∗}, d} and differential ideal 𝒦 ⊂ Φ^{∗} there is a d-invariant filtration

F^{p}𝒦 = image \{\underbrace{𝒦 ∧ … ∧ 𝒦}_{p} ∧ Φ^{∗} → Φ^{∗}\} .

1.4.1 The standard spectral sequence construction associated to a filtered complex then gives a spectral sequence E_{r}^{p,q}(𝒦) where

E_{∞} ⟹ H^{∗}(Φ^{∗})
E_{1}^{p,q}(𝒦) = H^{∗}(Gr^{p}𝒦)

where Gr^{p}𝒦 = F^{p}𝒦/F^{p+1}𝒦. This construction is functorial, and thus in our situation there is an induced mapping

F^{∗}: E_{r}^{p,q}(ℐ) ⟶ E_{r}^{p,q}(𝒥)

We note that

E_{1}^{0,∗}(ℐ) = H̄^{∗}

is the characteristic cohomology of the infinitely prolonged differential ideal ℐ. This is the object of our study.

Turning to the fibration 𝒩 → ℳ, we recall that there is a vector bundle, known informally as the “cohomology bundle”, ℋ^{q} → ℳ whose fiber over t ∈ ℳ is H_{DR}^{q}(N_{t}) and whose space of global sections is E_{1}^{0,q}(𝒥). Moreover, this bundle has an integrable (i.e., flat) connection

∇: ℋ^{q} → ℋ^{q} ⊗ Ω^{1}(ℳ),

known as the Gauss-Manin connection, whose horizontal sections are interpreted as geometrically displacing cycles. Now E_{1}^{1,q}(𝒥) ≅ ℋ^{q} ⊗ Ω^{1}(ℳ) and it is well known (and easy to see) that the differential in the spectral sequence

d_{1}: E_{1}^{0,q} → E_{1}^{1,q}

is the Gauss Manin connection. For φ ∈ ℋ^{q} this gives us that

(1)  \frac{d}{dt}(φ) = \frac{∂}{∂t} d_{1}φ.

This equation has the following interpretation: Given a point t_{0} ∈ ℳ and tangent vector ξ ∈ T_{t_{0}}ℳ, the left hand side of (1) is by definition the derivative at t_{0} of φ along any curve in ℳ with tangent vector ξ. By the interpretation of the Gauss-Manin connection, this is just ⟨∇φ, ξ⟩_{t_{0}}, which by our above remark is ξ d_{1}φ.

1.4.2 Putting this together we see that for φ ∈ H̄^{∗}

\frac{d}{dt}(f_{t}^{∗}φ) = \frac{∂}{∂t} d_{1}φ

where the ∂/∂t on the right hand side is interpreted as the normal vector along f_{t}(N) corresponding to the tangent vector ∂/∂t ∈ T_{t}ℳ.

Thus, the variation of characteristic cohomology is measured by the differential d_{1} in the spectral sequence associated to the differential ideal ℐ ⊂ Ω^{∗}(X).

Definition: We set E_{r}^{p,q} = E_{r}^{p,q}(ℐ) and call E_{r}^{p,q} the characteristic spectral sequence associated to the differential ideal ℐ. In particular,

E_{1}^{0,∗} = H̄^{∗}

is the characteristic cohomology associated to ℐ.

One point of this discussion is that once we agree to study the characteristic cohomology as an interesting object, then the characteristic spectral sequence naturally and inevitably enters. At a deeper level, we will see that our object of interest H̄^{∗} is in fact best studied indirectly in terms of the E_{r}^{p,∗} for p > 0. For example, the symbol of the exterior differential system will allow us to compute a first approximation to E_{1}^{p,∗} for p > 0.

In summary, we begin by studying the naïve characteristic cohomology H̄_{0}^{∗} associated to a differential ideal ℐ_{0} on a manifold X. The necessity of considering derivatives of all orders leads us to replace X_{0} and ℐ_{0} with the infinitely prolonged exterior differential system ℐ_{0} on X with its corresponding characteristic cohomology H̄^{∗}. But then variational considerations lead us to study the entire characteristic spectral sequence E_{r}^{p,q} of which H̄^{∗} = E_{1}^{0,∗}. We shall next isolate the essential algebraic construction which enables us to do this.

1.4.3 2. Frobenius Extensions

When one prolongs or differentiates a PDE system up to arbitrary order, what emerges is a filtered object together with differential operators having the property that on the associated graded the operators are algebraic – i.e., they are linear over the functions and depend only on the symbol on highest order terms of the original system. This led Spencer to the introduction of the “Spencer cohomology groups” associated to a symbol, and opened the way for a formal theory that greatly extended and clarified earlier work of Cartan and others. In the present problem of understanding the characteristic cohomology, a similar circumstance will obtain, albeit with one quite new and interesting twist. Underlying this development is a very simple and, we feel, very basic concept in what might be called exterior differential algebra. This concept is that of a Frobenius extension of differential graded algebra relative to a Pfaffian differential ideal. This section will be devoted to a discussion centering around that concept, and ultimately pointing the way to a new and somewhat stranger, although seemingly basic, algebraic construction associated to a symbol. 2.1 Structure equations of the prolongation tower

We shall arrive at the structure equations of the prolongation tower in a number of steps. To explain these we recall that each of the three constructions of Section 1.2

(i)  R ⊂ J^{1}(N, Y)  (classical PDE system)
(ii)  R ⊂ J^{1}(N, Z)  (PDE system associated to a fibered manifold)
(iii)  X_{1} ⊂ G_{n}(T X_{0})  (1st prolongation of an exterior differential system)

“looks” the same in local coordinates (x^{i}, u^{α}, p_{i}^{α}), except that the defining equations of X_{1} have a special form reflecting the fact that they arise from an exterior differential system, which is immaterial for the present discussion. What is different are the symmetry groups associated to three situations. We shall first give a local coordinate description of the structure equations of the prolongation tower that is invariant under the gauge group associated to construction (ii) above. Our second step will then be to reformulate these equations so as to be invariant under the full symmetry group of generalized contact transformations associated to construction (iii).

2.1.1

We begin by studying some constructions associated to the fibered manifold Z → N in the unconstrained case; i.e., when R = J^{1}(N, Z). Denote by J^{k}(N, Z) the manifold of k-jets j^{k}(s) of sections s: N → Z of the fibered manifold, and by J^{∞}(N, Z) = \underleftarrow{lim} J^{k}(N, Z) the space of jets of infinite order j^{∞}(s). In coordinates the projection J^{∞}(N, Z) → N is given by

π(x^{i}, u^{α}, …, p_{I}^{α}, …) = (x^{i})

We shall interpret the complete integrability of the infinitely prolonged contact system as giving a flat connection for the fibration J^{∞}(N, Z) → N.

For this we recall that functions G on J^{∞}(N, Z) are given by functions G(x^{i}, u^{α}, …, p_{I}^{α}), |I| ≦ k, on some finite jet manifold J^{k}(N, Z). Letting θ_{I}^{α} = dp_{I}^{α} − p_{Ii}^{α}dx^{i} denote the generators of the contact system on J^{∞}(N, Z) associated to our choice of local coordinates, we define operators d_{V}, d_{H} by

d_{V} G = ∑_{|I|≧0} \frac{∂G}{∂p_{I}^{α}} θ_{I}^{α}
dG = d_{V} G + d_{H}G

Explicitly,

d_{H}G = ∑_{i} D_{i}G dx^{i}

where

D_{i} = \frac{∂}{∂x^{i}} + ∑_{|I|≧0} p_{Ii}^{α} \frac{∂}{∂p_{I}^{α}}

is the operation of “total differentiation with respect to x^{i} ” – i.e.

(1)  \frac{∂}{∂x^{i}} (G(j^{∞}(s))) = (D_{i}G)(j^{∞}(s)).

The D_{i} are commuting vector fields on J^{∞}(N, Z) that satisfy

⟨θ_{I}^{α}, D_{i}⟩ = 0,

and thus they span an integrable distribution on J^{∞}(N, Z) that is horizontal relative to the projection π: J^{∞}(N, Z) → N. In fact, (1) expresses the fact that D_{i} is the horizontal lift of ∂/∂x^{i} relative to the integrable connection for J^{∞}(N, Z) → N.

2.1.2

Under the decomposition on J^{∞}(N, Z)

(2)  d = d_{V} + d_{H}

of the exterior derivative into vertical and horizontal components, we have

\{ \begin{matrix}d_{V}^{2} = d_{H}^{2} = 0 \\ d_{V} d_{H} + d_{H}d_{V} = 0 \end{matrix}

The decomposition (2) and integrable connection are gauge invariant, as follows e.g. from the coordinate free formulation

\{ \begin{matrix}d(j^{∞}(s)^{∗}G) & = j^{∞}(s)^{∗}(d_{H}G) \\ j^{∞}(s)^{∗}(d_{V} G) & = 0 \end{matrix}

of (1).

We note that

\{ \begin{matrix}d_{V} θ_{I}^{α} = 0 \\ d_{H}θ_{I}^{α} = −θ_{Ii}^{α} ∧ dx^{i}, \end{matrix}

and that trivially d_{V} (dx^{i}) = d_{H}(dx^{i}) = 0. The second of the bracketed equations is equivalent to

(3)  ℒ_{D_{i}}(θ_{I}^{α}) = θ_{Ii}^{α}

where ℒ_{D} is the Lie derivative operator associated to the vector field D on J^{∞}(N, Z). Suppose now that a constraint manifold R_{1} ⊂ J^{1}(N, Z) is defined by equations

F^{λ}(x^{i}, u^{α}, p_{i}^{α}) = 0.

In PDE theory one defines the k-th prolongation R_{k} ⊂ J^{k}(N, Z) of R to be the PDE system defined by the prolonged equations

D_{I}F^{λ} = 0, |I| ≦ k − 1

where D_{i_{1}...i_{ℓ}} = D_{i_{1}} … D_{i_{ℓ}}. Recall our assumption that each R_{k} is a manifold and that the projections R_{k} → R_{k−1} are surjective. We set R = \underleftarrow{lim} R_{k} and have a fibration diagram

R ⊂ J^{∞}(N, Z)
↓ ↓
N =  N

2.1.3 The pertinent observation is that the exterior differential system associated to the PDE system R_{k} is just the k-th prolongation, as defined in Section 1.2, of the exterior differential system associated to R_{1}. Thus, our terminology is unambiguous and we may say that the infinite prolongation is defined by the equations

D_{I}F^{λ} = 0.

Of course, we may also give the above construction in a coordinate free manner. For example, R_{k} is simply the set of k-jets of formal solutions to the given PDE system, where a k-jet of a solution is given by a smooth section of Z → N that satisfies the equation up to order k at one point of N. We have used the notation R_{k} and R to emphasize that this discussion is invariant under gauge transformation of local coordinates, but is not necessarily invariant under full contact transformations.

The equations defining R are invariant under the horizontal vector fields D_{i} (actually, all that is needed is that the defining ideal of R ⊂ J^{∞}(N, Z) be invariant), and thus along R the horizontal spaces are tangent to R. We conclude that the fibration R → N has a gauge invariant integrable connection, and we shall now denote by D_{i}, θ_{I}^{α}, d = d_{V} + d_{H} the restrictions to R of the quantities defined above on J^{∞}(N, Z).

The vertical forms θ_{I}^{α} are no longer linearly independent on X, but rather satisfy the relations that define T R ⊂ T J^{∞}(N, Z). To conveniently express these relations we recall our notation

Θ_{k} = ℐ_{k} ∩ Ω^{1}(R)
= span{θ_{I}^{α}: |I| ≦ k − 1}

The equations (1) imply that d_{H}(D_{I}F^{λ}) = 0 along R, i.e. in T (J^{∞}(N, Z))|_{R}. Thus, when restricted to the submanifold R we have

0 = d(D_{I}F^{λ}) = d_{V} (D_{I}F^{λ})

Taking I = ϕ, this gives

0 = \frac{∂F^{λ}}{∂p_{i}^{α}} θ_{i}^{α} + \frac{∂F^{λ}}{∂u^{α}} θ^{α}

⟹  \frac{∂F^{λ}}{∂p_{i}^{α}} θ_{i}^{α} ≡ 0 mod Θ_{1}

2.1.4 Taking I = {j}, we have

0 = d_{V} (D_{j}F^{λ})

= ℒ_{D_{j}}(d_{V} F^{λ})

≡ \frac{∂F^{λ}}{∂p_{i}^{α}} θ_{ij}^{α} mod Θ_{2}

by (3). Repeating this computation inductively, we obtain the defining relation

\frac{∂F^{λ}}{∂p_{i}^{α}} θ_{Ii}^{α} ≡ 0 mod Θ_{k+1}, |I| = k

for T R in T (J^{∞}(N, Z)). In conclusion: With the notations

Θ_{k} = ℐ_{k} ∩ Ω^{1}(X)
= ℐ^{1} ∩ Ω_{k}

we have the filtration

(4)  Θ_{1} ⊂ Θ_{2} ⊂ ⋯ Θ = ℐ^{1}

of the 1-forms in ℐ. The subspaces Θ_{k} satisfy the structure equations

(5)  dΘ_{k} ≡ 0 mod {Θ_{k+1}}

and

Θ_{k}/Θ_{k−1} = span{θ_{I}^{α}: |I| = k − 1}

where the θ_{I}^{α} satisfy the symbol relations

(6)  (\frac{∂F^{λ}}{∂p_{i}^{α}} ) θ_{Ii}^{α} ≡ 0 mod Θ_{k},  |I| = k − 1.

We have observed that a gauge transformation preserves the filtration (4) and preserves the complement of Θ in Ω^{1}(R) given by the horizontal 1-forms for the flat connection

2.1.5 in R → N. Suppose now that we consider the original PDE system as an exterior differential system ℐ_{1} on a manifold X_{1}. Of course, R_{1} and X_{1} are diffeomorphic as manifolds but the symmetry group of X_{1} may be strictly larger than that of R_{1}, hence we use the different notations. As noted above, the manifold R_{k} underlying the k-th prolongation of R_{1} as a PDE system is diffeomorphic to the manifold X_{k} underlying the k-th prolongation ℐ_{k} of the exterior differential system ℐ_{1}. We have pointed out that the exterior differential system corresponding to R_{k} is just ℐ_{k} on X_{k}, but again as before the symmetry group of the latter may be larger. Passing to jets of infinite order, an automorphism of the prolongation tower {R_{k}} of R_{1} will induce an automorphism of the prolongation tower {X_{k}}, and hence an automorphism of X preserving ℐ. But an automorphism T of ℐ need not preserve the complement of Θ in Ω^{1}(X); i.e., in coordinates we will only have

T^{∗}dx^{i} ≡ A_{j}^{i}dx^{j} mod Θ
T^{∗}θ_{I}^{α} ≡ 0 mod Θ_{k} for |I| ≦ k − 1.

Nevertheless, the relations (4)–(6) are formulated in a manner invariant under such transformations, hence under Aut(ℐ), and are therefore valid on X.

In summary, the relations (4)–(6) are not only gauge invariant, but are invariant under the full symmetry group of the exterior differential system and hence have meaning on (X, ℐ).

2.1.6 2.2 The symbol relations

In the preceding section we have arrived at the structure equations

(1)  \{ \begin{matrix}(i) & dΘ_{k} ≡ 0 mod \{Θ_{k+1}\}, \\ (ii) & \frac{∂F^{λ}}{∂p_{i}^{α}} θ_{Ii}^{α} ≡ 0 mod Θ_{k+1}, |I| = k \end{matrix}

of the prolongation tower. The first equations are formulated in a coordinate free manner that is invariant under the full symmetry group of the exterior differential system. In this section we will give a similar intrinsic formulation of (ii).

For this we recall the construction of G_{n,∞}(X_{0}) = \underleftarrow{lim} G_{n,k}(X_{0}), whose points are jets of infinite order of n-dimensional submanifolds of X_{0}. For all k including k = ∞ there are fibrations

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  π G_{n,k}(X_{0})  X^{0} G^{n,1}(X^{0})

and over G_{n,1}(X_{0}) = G_{n}(T X_{0}) there are defined the tautological vector bundles

\{ \begin{matrix}\underline{E} & ⊂ π^{∗}(T X_{0}) \\ \underline{W} & = π^{∗}(T X_{0})/\underline{E} \end{matrix}

In terms of the local coordinates used in Section 1.3, \underline{E} has a framing ∂/∂x^{1}, …, ∂/∂x^{n} and \underline{W} has a framing ∂/∂u^{1}, …, ∂/∂u^{s}. We shall continue to denote by \underline{E}, \underline{W} the pullbacks of these bundles to any G_{n,k}(X_{0}) or to G_{n,∞}(X_{0}).

The first basic observation is that over G_{n,k}(X_{0}) we have exact sequences

0 → \underline{W} ⊗ S^{k}\underline{E}^{∗} → T G_{n,k}(X_{0}) → T G_{n,k−1}(X_{0}) → 0

and dually

0 → T^{∗}G_{n,k−1}(X_{0}) → T^{∗}G_{n,k}(X_{0}) → \underline{W}^{∗} ⊗ S^{k}\underline{E} → 0

where S^{k} denotes the k-th symmetric product and as usual we are omitting the pullback notation. We shall always lift these sequences up to G_{n,∞}(X_{0}), and when we do so and use the local coordinates introduced above the 1-forms θ_{I}^{α}, |I| = k, give a framing for

2.2.1 T^{∗}G_{n,k}(X_{0})/T^{∗}G_{n,k−1}(X_{0}) ≅ \underline{W}^{∗} ⊗ S^{k}\underline{E}. For a function G(x^{i}, u^{α}, …, p_{I}^{α}), |I| ≦ k, on G_{n,k}(X_{0}) the differential dG is an element of T^{∗}G_{n,k}(X_{0}) and the map T^{∗}G_{n,k}(X_{0}) → \underline{W}^{∗} ⊗ S^{k}\underline{E} → 0 is given by

dG → ∑_{|I|=k} \frac{∂G}{∂p_{I}^{α}} θ_{I}^{α}.

The existence of the above sequences is just a notationally complicated way of saying that the highest order derivatives are tensors. The one slight subtlety is that we shall use θ_{I}^{α} and not dp_{I}^{α} as a coframing of T^{∗}G_{n,k}(X_{0})/T^{∗}G_{n,k−1}(X_{0}); i.e., we choose our coframing to lie in the contact ideal.

It is when we restrict the second sequence above to the prolongation tower of ℐ_{0} that the symbol relations appear. More precisely, the pulling back and restriction of differentials associated to the diagram

X_{k} ⊂ G_{n,k}(X_{0})
↓ ↓
X_{k−1} ⊂ G_{n,k−1}(X_{0})

induces on the infinite prolongation X a diagram

T^{∗}G_{n,k}(X_{0})/T^{∗}G_{n,k−1}(X_{0}) → T^{∗}X_{k}/T^{∗}X_{k−1} → 0
↓^{≈}  ↓^{≈}
0 → \underline{B}_{k−1} →  \underline{W}^{∗} ⊗ S^{k}\underline{E}  →  \underline{M}_{k}  → 0

where \underline{B}_{k−1}, \underline{M}_{k} are notations. The vector bundle \underline{M}_{k} is spanned by the restrictions of the θ_{I}^{α}, |I| = k, to X and the fact that these forms are subject to the symbol relations given by equations (ii) in (1) is expressed by saying that \underline{B}_{k−1} ⊂ \underline{W}^{∗} ⊗ S^{k}\underline{E} is spanned by the forms

(2)  (\frac{∂F^{λ}}{∂p_{i}^{α}} ) θ_{Ji}^{α},  |J| = k − 1.

2.2.2 2.3 Frobenius extensions

Most of the calculations in the theory of exterior differential systems have an algebraic, almost algorithmic character; the fact that we are working on a manifold is not the central point, and it may even be distracting. Moreover, many essential points of the theory such as a prolongation and involution are most clearly formulated algebraically, even through they are motivated geometrically. In this and the next section we shall isolate in a purely algebraic formulation what we feel are the essential general aspects of the study of characteristic cohomology of an exterior differential system.

To motivate this discussion, we observe that localizing (i.e., sheafifying) our construction at a point of X, we arrive at the data

(1)  {Ω^{∗}, ℐ, Ω_{k}, d}

where

(i) {Ω^{∗}, d} is a differential graded algebra (DGA) (ii) ℐ is a Pfaffian differential ideal (iii) Ω_{k}^{∗} is an increasing filtration of Ω^{∗} by subalgebras such that each ℐ_{k} = Ω_{k}^{∗} ∩ ℐ is a

Pfaffian differential ideal.

There are a number of axioms that the data (1) will need to satisfy, of which the essential one is this: Setting Θ_{k} = ℐ_{k}^{1} and denoting by {Θ_{k+1}} the algebraic ideal in Ω_{k+1}^{∗} generated by Θ_{k+1}, we have

dΘ_{k} ≡ 0 mod {Θ_{k+1}}.

Intuitively, the extensions of DGA’s

⋯ ⊂ Ω_{k} ⊂ Ω_{k+1} ⊂ ⋯

are obtained by adjoining 1-forms such that ℐ_{k} satisfies the Frobenius integrability condition relative to ℐ_{k+1}.

We formalize this concept with the following

2.3.1 Definition: We consider the data {Λ^{∗}, 𝒥, d} where Λ^{∗} is a differential graded algebra and 𝒥 ⊂ Λ^{∗} is a Pfaffian differential ideal.

(a) The above data is said to be regular in case Λ^{0} is a regular local ring and Λ^{∗}, ℐ are free modules over Λ^{0}.

(b) Given an inclusion of regular data

(2)  {Λ^{∗}, 𝒥, d} ⊂ {Λ̃^{∗}, 𝒥̃, d},

such that Λ^{0} and Λ̄^{0} have the same residue class field, we say that (2) is a Frobenius extension if

(3)  \{ \begin{matrix}(i) & Λ̃^{∗} is generated algebraically by Λ^{∗} and 𝒥̃. \\ (ii) & 𝒥 = Λ^{∗} ∩ 𝒥̃, \\ & and setting Ψ = 𝒥^{1} and Ψ̃ = 𝒥̃^{1} \\ (iii) & dΨ ≡ 0 mod \{Ψ̃\}. \end{matrix}

(c) The data (1) above where each inclusion {Ω_{k}^{∗}, ℐ_{k}, d} ⊂ {Ω_{k+1}^{∗}, ℐ_{k+1}, d} is a Frobenius extension and where Ω^{∗} = ∪_{k≧0} Ω_{k}^{∗}, ℐ = ∪_{k≧0} ℐ_{k} will be called a Frobenius tower.

As remarked above, intuitively the extension (2) is Frobenius if Λ̃^{∗} is obtained from Λ^{∗} by adjoining 1-forms (condition (i) above) such that in so doing 𝒥 is “closed up” relative to the Frobenius integrability condition ((iii) above), and finally if this extension is non-trivial in the sense that we do not obtain 𝒥̃ by adding 1-forms already in Λ^{∗} (condition (ii)). We may also speak of other properties, such as minimal and universal Frobenius extensions. We shall not enter into such a discussion, but rather we shall explain two conditions on {Λ, 𝒥, d} that will guarantee that a Frobenius extension exists – this construction will clearly exhibit properties of minimality and universality. These two conditions are motivated by circumstances prevailing in the construction of the prolongation tower of an exterior differential system, which we shall refer to as the geometric situation.

Construction of Frobenius extensions: One may ask when regular data {Ω_{1}^{∗}, ℐ_{1}, d} admits a Frobenius extension that is in some sense minimal and universal. We are using the notation Ω_{1}^{∗}, ℐ_{1} rather than Λ^{∗}, 𝒥 to suggest the initial step in the construction of a

2.3.2 Frobenius tower. We shall give two conditions on this data which mirror the geometric situation and which guarantee the existence of such a Frobenius extension {Ω_{2}^{∗}, ℐ_{2}, d}. Attempting to continue the process, the first of these conditions will automatically be satisfied for the Frobenius extension {Ω_{2}^{∗}, ℐ_{2}, d}. The second condition, which mirrors the surjectivity of X_{2} → X_{1}, will have to be assumed at each stage in the construction of the Frobenius tower. However, the first condition will allow us to define what it means for {Ω_{1}^{∗}, ℐ_{1}, d} to be involutive, and then in the involutive case the second condition will also be satisfied at each stage in the construction of the Frobenius tower.

We set ℱ_{1} = Ω_{1}^{0} with residue field 𝔽 and Θ_{1} = ℐ_{1}^{1}. The first condition on the regular data {Ω_{1}^{∗}, ℐ_{1}, d} is: There exists a subspace Ω_{0}^{1} ⊂ Ω_{1}^{1} such that

(3)  dΘ_{1} ≡ Ω_{1}^{1}/Ω_{0}^{1} ⊗ Ω_{0}^{1} mod {Θ_{1}}.

In the geometric situation we may take Ω_{0}^{1} = π^{∗}Ω^{1}(X_{0}) where π: X_{1} → X_{0} is the projection; the condition (3) follows from the structure equations discussed in Section 2.1 above.

With this assumption there exist free ℱ_{1}-module generators θ^{α} for Θ_{1} and ω^{i} for Ω_{0}^{1} such that

(4)  dθ^{α} ≡ −π_{i}^{α} ∧ ω^{i} mod {Θ_{1}}

for some elements π_{i}^{α} ∈ Ω_{1}^{1}. These π_{i}^{α} are only well-defined modulo the ℱ_{1}-span of Ω_{0}^{1} and Θ_{1}. Moreover, they are not linearly independent but are subject to a minimal set of symbol relations

b_{α}^{λi}π_{i}^{α} ≡ 0 mod {Ω_{0}^{1}, Θ_{1}}

where b_{α}^{λi} ∈ ℱ_{1}. The non-uniqueness of the π_{i}^{α} is expressed by saying that they are welldefined modulo Θ_{1} up to a substitution

(5)  π_{i}^{α} → π_{i}^{α} + p_{ij}^{α}ω^{j},  p_{ij}^{α} = p_{ji}^{α} ∈ ℱ_{1}.

The symmetry condition is necessary in order that the equations (4) remain valid. Our second condition is that: There exists a substitution (5) such that the refined symbol relations

b_{α}^{λi}π_{i}^{α} ≡ 0 mod Θ_{1}

2.3.3 are satisfied.

As explained in [2], Chapter IV, in the geometric case this condition is equivalent to the assumption that X_{2} → X_{1} be surjective. Of course, it can be formulated intrinsically in terms of the data {Ω_{1}^{∗}, Ω_{0}^{1}, ℐ_{1}, d} but that would take us too far afield here.

Once we impose the refined symbol relations, the π_{i}^{α} are unique up to a substitution (5) where

(6)  b_{α}^{λi}p_{ij}^{α} = 0,  p_{ij}^{α} = p_{ji}^{α}.

We then define

ℱ_{2} = ℱ_{1}[p_{ij}^{α}]

where the p_{ij}^{α} are indeterminates over 𝔽 subject to (6), and we then define Ω_{2}^{∗} to be the DGA generated by Ω_{1}^{∗} and the p_{ij}^{α} and their differentials dp_{ij}^{α}, subject of course to the relations obtained by differentiating (6). Finally, we define ℐ_{2} ⊂ Ω_{2}^{∗} to be the Pfaffian differential ideal generated as a differential ideal by ℐ_{1} and the 1-forms

θ_{i}^{α} = π_{i}^{α} + p_{ij}^{α}ω^{j} ∈ Ω_{2}^{1}.

The equation (4) now reads

dθ^{α} ≡ −θ_{i}^{α} ∧ ω^{i} mod {Θ_{1}},

which is the Frobenius extension condition (iii). The other conditions (i) and (ii) are clearly satisfied, and thus {Ω_{2}^{∗}, ℐ_{2}, d} is a Frobenius extension of {Ω_{1}^{∗}, ℐ_{1}, d}.

We now want to show that condition (3) is inherited by {Ω_{2}^{∗}, ℐ_{2}, d}. Exterior differentiation of the preceding equation gives

0 ≡ −dθ_{i}^{α} ∧ ω^{i} mod {Θ_{2}}.

Applying the Cartan lemma, we obtain

(7)  dθ_{i}^{α} ≡ −π_{ij}^{α} ∧ ω^{j} mod {Θ_{2}}

where

π_{ij}^{α} ≡ π_{ji}^{α} mod Θ_{2}

2.3.4 As before, we may refine this last relation to

π_{ij}^{α} = π_{ji}^{α}.

Equation (7) is just the first condition (3) for {Ω_{2}^{∗}, ℐ_{2}, d}, where we keep the same subspace Ω_{0}^{1}.

We now investigate the second condition for {Ω_{2}^{∗}, ℐ_{2}, d}. Exterior differentiation of

b_{α}^{λi}θ_{i}^{α} ≡ 0 mod Θ_{1}

yields, after some fiddling and application of the Cartan Lemma, the symbol relations

b_{α}^{λi}π_{ij}^{α} ≡ 0 mod Ω_{0}^{1}, Θ_{2}

The π_{ij}^{α} satisfying (7) and the symmetry condition are uniquely defined up to a substitution

(8)  π_{ij}^{α} → π_{ij}^{α} + p_{ijk}^{α}ω^{k}

where p_{ijk}^{α} ∈ ℱ_{2} and is symmetric in i, j, k. We must then assume that we may determine p_{ijk}^{α} so as to have the refined symbol relations

b_{α}^{λi}π_{ij}^{α} ≡ 0 mod Θ_{2}.

In the geometric case this assumption is equivalent to X_{3} → X_{2} being surjective. Once it is satisfied, the π_{ij}^{α} satisfying (7) and the refined symbol relations are uniquely determined up to a substitution (8) where

b_{α}^{λi}p_{ijk}^{α} = 0

We set ℱ_{3} = ℱ_{2}[p_{ijk}^{α}] where the p_{ijk}^{α} are indeterminates over 𝔽 subject to these linear equations plus the symmetry in i, j, k, and proceed as before to define {Ω_{3}^{∗}, ℐ_{3}, d} where ℐ_{3} is the Pfaffian differential ideal generated by

θ_{ij}^{α} = π_{ij}^{α} + p_{ijk}^{α}ω^{k}.

We may continue the argument, constructing a Frobenius tower under the assumptions that the initial data satisfies the condition (3) and that, at each stage, we may refine the symbol relations

b_{α}^{λi}θ_{Ii}^{α} ≡ 0 mod Ω_{0}^{1}, Θ_{k}, |I| = k

2.3.5 to

b_{α}^{λi}θ_{Ii}^{α} ≡ 0 mod Θ_{k}, |I| = k

In so doing the only technical tool required for the inductive step is the Cartan-Poincaré lemma on page 321 in [2].

Comparing this discussion with the discussion of the structure equations in Section 2.1 and symbol relations in Section 2.2 we see that (3) is satisfied in the geometric case and that moreover the refined symbol relations are satisfied under the assumption that the projections X_{k+1} → X_{k} are all submersive.

It is important to remark that there is a condition on the symbol matrices

B(ξ) = ∥b_{α}^{λi}ξ_{i}∥

that will guarantee that the refined symbol relations are satisfied for all k ≧ 2, provided only that they are satisfied for k = 1. This is the condition that the symbol be 2-acyclic as defined in [ ], chapter VIII. In the geometric case, the involutivity of the symbol matrices ∥∂F^{λ}/∂p_{i}^{α}ξ_{i}∥ will guarantee that the maps X_{k+1} → X_{k} are submersive for all k ≧ 2 provided that this is the case for k = 1. We will then say that the data {Ω_{1}^{∗}, ℐ_{1}, d} is involutive. As explained in Chapter VI of [2], at least in theory we may always reduce to the involutive case.

In summary, if we begin with a regular DGA {Ω_{1}^{∗}, ℐ_{1}, d} which satisfies condition (3) above, for which the refined symbol relations hold and for which the symbol is involutive, then we may construct a Frobenius tower {Ω^{∗}, Ω_{k}^{∗}, ℐ, d}. In the next section we will investigate the behavior of this construction under automorphisms (which need not preserve Ω_{0}^{1} ⊂ Ω_{1}^{1}), and shall show that the Frobenius tower construction leads to a weight filtration on the associated graded to the filtration of Ω^{∗} defined by the powers of ℐ. This weight filtration will have a remarkable property which in practice allows us to get a first approximation to our object of interest H̄^{∗} = H{Ω̄^{∗}, d̄} purely from properties of the symbol.

2.3.6 2.4 The weight filtration

Associated to the data {Ω^{∗}, ℐ, d} we have defined the filtration

F^{p}Ω^{∗} = image of \{\underbrace{ℐ ∧ … ∧ ℐ}_{p} ∧ Ω^{∗} → Ω^{∗}\} ,

to which there is the associated graded complex

F^{p}Ω^{∗}/F^{p+1}Ω^{∗} =: Ω̄^{p,∗}

= ⊕_{q} Ω̄^{p,q}.

Suppose now that we have a Frobenius tower {Ω^{∗}, ℐ, Ω_{k}^{∗}, d}. We will use the increasing filtration of Ω^{∗} given by the Ω_{k}^{∗} to define an intrinsic weight filtration on each of the complexes {Ω̄^{p,∗}, d̄}, one which has the remarkable property that on its associated graded the induced differential is linear over the functions, a property traceable to the Frobenius extension condition. Definition/Proposition: We consider all functions

w: Ω^{∗} → ℤ

which satisfy (i) w(fφ) = w(φ) for f ∈ ℱ, f ≠ 0 (ii) w(φ) ≦ k for φ ∈ Θ_{k+1}, and if equality holds then φ ∉ Θ_{k} (iii) w(φ∧η) ≦ w(φ) + w(η), and (iv) w(φ + η) ≦ max(w(φ), w(η)). We set

wt φ = max_{w} w(φ)

where the maximum is over all functions w satisfying (i)–(iv) above. Then, for p > 0 “wt” is well-defined on the associated graded complexes Ω̄^{p,∗}, and if we set

F̄_{k}Ω̄^{p,∗} = {φ ∈ Ω̄^{p,∗} : wt(φ) ≦ k}
Ω̄{p,∗ \atop k} = F̄_{k}Ω̄^{p,∗}/F̄_{k−1}Ω̄^{p,∗}

= ⊕_{q} Ω̄{p,q \atop k}

2.4.1 we obtain a \underline{weight filtration} \{F̄_{k}\} that is stable under d̄ and has the property that the induced differential:

(1)  δ : Ω̄{p,q \atop k} → Ω̄{p,q+1 \atop k}is ℱ-\underline{linear} (p > 0).

We shall give the actual construction of the weight filtration under the assumption that {Ω_{1}^{∗}, ℐ_{1}, d} satisfies (3) in the preceding section and that the refined symbol relations hold at each step. As noted above, these assumptions are both satisfied in the involutive geometric case. Before giving the construction we remark that the crucial property is (1). By means of this property the groups E_{1}^{p,∗} (for p > 0) will themselves be the abutment of a spectral sequence, the one coming from the weight filtration whose E_{1}-term Ē_{1}^{k,∗} will be calculated purely algebraically. It is here that the symbol and Spencer cohomology of the exterior differential system enter the picture, in a more or less standard setting for p = 1 but in a new and algebraically rather subtle way for p ≧ 2.

Construction of the weight filtration. We will follow the notations in the preceding section, especially those used in the construction of Frobenius extensions. Accordingly, we may find a spanning set of forms

(2)  ω^{i}, θ_{I}^{α}

for Ω^{1} with the properties that

i) the ω^{i} ∈ Ω^{1} ∖ Θ ii) θ_{I}^{α} ∈ Θ_{k+1} for |I| ≦ k iii) b_{α}^{λi}θ_{Ii}^{α} ≡ 0 mod Θ_{k} for |I| = k.

The relations (iii) are a defining set of relations among the forms ω^{i}, θ_{K}^{α}.

Now properties (i)–(iii) imply that

wt ω^{i} = −1
wt θ_{I}^{α} = |I| (provided that θ_{I}^{α} ≢ 0 mod Θ_{|I|}).

2.4.2 These properties, together with (iii), give

(3)  wt(θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ ω^{j_{1}} ∧ … ∧ ω^{j_{q}}) = ∑_{ν} |I_{ν}| − q

provided of course that the expression on the left is not zero. Finally, (iv) allows us to define wt φ for a general form (repeated indices are summed)

φ = f_{α_{1}...α_{p}J}^{I_{1}...I_{p}}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ ω^{J}

We must check how “wt” behaves when we make an admissible change of a spanning set of forms (2), and then we must check how “wt” interacts with exterior differentiation. For the former, the ω^{i} are determined up to a transformation

ω^{i} → A_{j}^{i}ω^{j} + (terms in Θ), det ∥A_{j}^{i}∥ ≠ 0,

and recalling that θ_{I}^{α} ∈ Θ_{|I|+1} the θ_{I}^{α} are determined up to a transformation

θ_{I}^{α} → ∑_{|J|=|I|} A_{Iβ}^{αJ}θ_{J}^{β} + (terms in Θ_{|I|}), det ∥A_{Iβ}^{αI}∥ ≠ 0.

It follows that (3) is well-defined provided that we consider θ_{I_{1}}^{α_{1}} ∧ … ∧θ_{I_{p}}^{α_{p}} ∧ω^{j_{1}} ∧ … ∧ω^{j_{q}} as an element in F^{p}Ω^{∗}/F^{p+1}Ω^{∗}. Put another way, if we assume that the matrices ∥A_{j}^{i}∥ and ∥A_{Iβ}^{αJ}∥ are each the identity, then under the above transformation we have

θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ ω^{J} → θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ ω^{J} + (terms in \underbrace{ℐ ∧ … ∧ ℐ}_{p+1} ∧ Ω^{∗})

+ (terms spanned by forms θα̃_{1} ; Ĩ_{1} ∧ … ∧ θα̃_{p} ; Ĩ_{p} ∧ ωJ̃ where all

|Ĩ_{ν}| ≦ |I_{ν}|, |J| = |J̃| and some |Ĩ_{ν}| < |I_{ν}|)

The conclusion is that “wt” is well-defined on Ω̄^{p,∗} = F^{p}Ω^{∗}/F^{p+1}Ω^{∗}.

We note that in Ω̄{1,∗ \atop k} we have the defining relations

(4)  b_{α}^{λi}θ_{Ii}^{α} = 0, |I| = k − 1.

More generally, in Ω̄{p,∗ \atop k} the relations among the spanning set of forms

θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ ω^{J},  ∑_{ν} |I_{ν}| − |J| = k

2.4.3 are linearly generated by the relations (4).

Finally, it remains to investigate how d̄= d mod ℐ interacts with the weight filtration on Ω̄^{p,∗}. For this we recall the structure equations for the spanning set of forms (1)

\{ \begin{matrix}dω^{i} & ≡ \frac{1}{2}c_{jk}^{i}ω^{j} ∧ ω^{i} mod \{Θ\} \\ dθ_{I}^{α} & ≡ −θ_{Ii}^{α} ∧ ω^{i} mod \{Θ_{k}\}, k = |I| + 1 \end{matrix}

From these it follows that

(5)  \{ \begin{matrix}d̄ω^{i} & ≡ 0 modulo (ℐ + terms of lower weight) \\ d̄θ_{I}^{α} & ≡ −θ_{Ii}^{α} ∧ ω^{i} modulo (terms of lower weight). \end{matrix}

Moreover, for a function f

df = ∑ f_{i}ω^{i} + (terms in Θ)
⟹  d̄f = ∑ f_{i}ω^{i} has weight −1.

Combining this with (5) we see that in the associated graded spaces to the weight filtration Ω̄{p,∗ \atop k}.

\begin{aligned}δ(f_{α_{1}...α_{p}J}^{I_{1}...I_{p}}θ_{I_{1}}^{α_{1}} & ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ ω^{J}) = \\ & (−1)^{p} ∑_{ν,i} f_{α_{1}...α_{p}J}^{I_{1}...I_{p}}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{ν}i}^{α_{ν}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ ω^{i} ∧ ω^{J}. \end{aligned} (6)

This establishes the property (1), and moreover (6) gives us a formula for δ on Ω̄{p,∗ \atop k} that will be used below.

We note in closing the trivial but important point that there is no alternation of signs with the index ν on the right hand side of (6). If we let S = 𝔽[e_{1}, …, e_{n}] and define an S-module action by

e_{i} · θ_{I}^{α} = θ_{Ii}^{α} ⊗ ω^{i}

and extend this to products θ_{I_{1}}^{α_{1}} ∧ … ∧θ_{I_{p}}^{α_{p}} ⊗ ω^{J} to act by the Leibnitz rule (with all plus signs) on θ_{I_{1}}^{α_{1}} ∧ … ∧θ_{I_{p}}^{α_{p}}, then the right hand side of (6) is essentially a Koszul boundary operator for this module structure. We shall explore this in detail in the following sections.

2.4.4 2.5 A variant of Spencer cohomology

We continue with our consideration of a Frobenius tower {Ω^{∗}, ℐ, Ω_{k}^{∗}, d} of Pfaffian differential ideals. We shall assume that this tower arises by an inductive construction, starting with {Ω_{1}^{∗}, ℐ_{1}, d}, where at each step the two conditions consisting of (3) and the refined symbol relations discussed in Section 2.3 are satisfied. As explained in the preceding section, we may construct the weight filtration on the associated graded complexes Ω̄^{p,∗} for p > 0, and we may then consider the induced differential

δ: Ω̄{p,q \atop k} → Ω̄{p,q+1 \atop k}

on the associated graded complexes to the weight filtration. From equations (1) and (6) in the preceding section we see that δ treats the functions ℱ = Ω^{0} as constants and the forms in Ω_{0}^{1} as being closed. Thus, the cohomology of the complex \{Ω̄{p,∗ \atop k}, δ\} is for p > 0 “purely algebraic”, and as we shall now see it is a variant of Spencer cohomology. Actually, it will turn out to be a variant in two ways. The first essentially trivial one is that a Poincaré-type duality will intervene. More significantly, for p = 1 we will find the usual Spencer or Koszul cohomology associated to a module M, whereas for p ≧ 2 we will obtain Spencer cohomology associated to Λ_{Δ}^{p}M where “Λ_{Δ}^{p}” is an apparently new and to us quite interesting multilinear algebra construction. Review of Spencer-Koszul cohomology

Let E be a vector space over a field 𝔽 of characteristic zero, and set

S^{k}E = Sym^{k}E = k^{th} symmetric product of E
S = SE = ⊕_{k≧0} S^{k}E = polynomial algebra on E.

Let N be a graded (but not necessarily finitely generated) graded S-module and recall the standard definition of Koszul homology: Setting

C_{p}(N) = N ⊗_{𝔽} Λ^{p}E

with action by S given by f(n ⊗ ω) = fn ⊗ ω where f ∈ S, n ∈ N and ω ∈ Λ^{p}E, one defines the S-linear mapping

∂: C_{p}(N) → C_{p−1}(N)

2.5.1 by the usual formula

∂(n ⊗ e_{1} ∧ … ∧ e_{p}) = ∑_{i}(−1)^{i}e_{i} · n ⊗ e_{1} ∧ … ∧ ê_{i} ∧ … ∧ e_{p}

where n ∈ N and e_{i} ∈ E. Then ∂^{2} = 0 and the resulting Koszul homology will be denoted by H_{p}(N). Clearly H_{∗}(N) = ⊕H_{p}(N) is again a graded S-module.

Dually we set

C^{q}(N) = N ⊗ Λ^{q}E^{∗}

and define

δ: C^{q}(N) → C^{q+1}(N)

by

δω(e_{1}, …, e_{q+1}) = ∑_{i}(−1)^{i}e_{i} · ω(e_{1}, …, ê_{i}, …, e_{q+1})

where ω ∈ N ⊗ Λ^{q}E^{∗} ≅ N ⊗ (Λ^{q}E)^{∗} and the e_{i} ∈ E. Alternatively, we let Id ∈ E ⊗ E^{∗} ≅ Hom(E, E) be the identity and define

Id: N ⊗ Λ^{q}E^{∗} → N ⊗ Λ^{q+1}E^{∗}

by

Id(n ⊗ e_{i_{1}}^{∗} ∧ … ∧ e_{i_{1}}^{∗}) = ∑ e_{i}n ⊗ e_{i}^{∗} ∧ e_{i_{1}}^{∗} ∧ … ∧ e_{i_{1}}^{∗}

where n ∈ N and e_{i} ∈ E is a basis with dual basis e_{i}^{∗} ∈ E^{∗} – thus Id = ∑_{i} e_{i} ⊗ e_{i}^{∗}. Then it is easy to verify that

(1)  δω = −Id ω

where ω ∈ C^{q}(N). The reason for the minus sign will appear below.

To relate these two constructions, we suppose that dim E = n and choose a non-zero volume from Ω ∈ Λ^{n}E. Contraction with Ω gives an isomorphism

i(Ω) · Λ^{q}E^{∗} → Λ^{n−q}E

2.5.2 and a commutative diagram

N ⊗ Λ^{n−p}E^{∗} ⟶^{δ} N ⊗ Λ^{n−p+1}E^{∗}
↓^{≀} ↓^{≀}
N ⊗ Λ^{p}E ⟶^{−∂} N ⊗ Λ^{p−1}E

leading to an isomorphism

i(Ω): H^{n−p}(N) →^{∼} H_{p}(N).

A different choice of Ω alters this isomorphism by a non-zero element in 𝔽.

We now want to relate this construction to the formula (6) from the preceding section when p = 1. For this we use the notation

𝒰. = 𝒰 ⊗_{𝔽} S

for the free S-module corresponding to a vector space 𝒰. Below we shall take 𝒰, E to be the fibers W^{∗}, E of the vector bundles \underline{W}^{∗}, \underline{E} introduced in Section 2.2 above. For the moment we assume given vector spaces 𝒰_{1}, 𝒰_{0} = 𝒰 and a linear mapping

(2)  b: 𝒰_{1} → 𝒰_{0} ⊗ E,

which we shall refer to as a symbol mapping. In practice, the vector spaces 𝒰_{1}, 𝒰_{0}, E will have bases u^{λ}, u^{α}, e_{i} such that

b(u^{λ}) = b_{α}^{λi}u^{α} ⊗ e_{i}

where the b_{α}^{λi} are the quantities introduced in the preceding section. Associated to (2) is an S-linear mapping of free modules

𝒰_{1,.} ⟶^{b} 𝒰_{0,.}

where

b(u^{λ}e_{I}) = b_{α}^{λi}u^{α}e_{i}e_{I}

(we shall omit the ⊗ symbol), and we denote by M the cokernel so that we have

(3)  𝒰_{1,.} ⟶^{b} 𝒰_{0,.} → M → 0.

2.5.3 The 𝔽-basis u^{α}e_{I} of 𝒰_{0, .} projects to an 𝔽-spanning set of vectors [u^{α}e_{I}] ∈ M subject to the 𝔽-spanning set of relations

(4)  b_{α}^{λi}[u^{α}e_{i}e_{I}] = 0.

M is usually called the graded S-module associated to the symbol b and b(𝒰_{1,.}) ⊂ 𝒰_{0,.} simply the symbol module.

We now denote by dx^{i} ∈ E^{∗} the basis dual to e_{i} and consider the spanning vectors

[u^{α}e_{I}] ⊗ dx^{J} ∈ C^{q}(M), |J| = q.

Then by our formula (1) above

(5)  δ[u^{α}e_{I}] ⊗ dx^{J} = −[u^{α}e_{i}e_{I}] ⊗ dx^{i} ∧ dx^{J}.

Comparing with the preceding section, we find that the relations (4) are exactly those imposed on the θ_{I}^{α} considered in the associated graded to the weight filtration, and (5) is exactly the formula (6) in that section for δ(θ_{I}^{α}∧ω^{J}) under the correspondence

(6)  \{ \begin{matrix}[u^{α}e_{I}] & ⟷ θ_{I}^{α} \\ ω^{i} & ⟷ dx^{i}. \end{matrix}

To be able to reduce the cohomology of \{Ω̄{1,∗ \atop k}, δ\} to Spencer cohomology we need to make two further remarks. The first is that for p > 0 we may consider the cohomology of \{Ω̄{p,∗ \atop k}, δ\} as a module over the local ring ℱ_{k} = Ω_{k}^{0}; this is because δ is linear over the functions. Denoting by m_{k} ⊂ ℱ_{k} the maximal ideal, there is a common residue field 𝔽 = ℱ_{k}/m_{k} for all k. It is the reduction modulo m_{k} of the cohomology of \{Ω̄{p,∗ \atop k}, δ\} that will be a Spencer-type cohomology. By Nakayama’s lemma we will be able to use results about this cohomology to infer information about the cohomology of \{Ω̄{p,∗ \atop k}, δ\}.

The other remark is that even after we reduce modulo m_{k} the θ_{I}^{α} do not have values in the k-th graded piece M_{k} of M, but rather have values in M_{k} ⊗ T^{∗} for some vector space T, reflecting the fact that the θ_{I}^{α} are differential forms. (In the geometric case, T^{∗} = T_{p}^{∗}X_{k}/T_{p}^{∗}X_{k−1}.) With these two points understood we may summarize our discussion as follows:

2.5.4 (7) For p = 1, we consider the reduction modulo m_{k} of the complex \{Ω̄{1,∗ \atop k}, δ\}. This is

then isomorphic to the k-th graded piece of {C^{∗}(M) ⊗ T^{∗}, δ} where M is the graded module associated to the symbol of {Ω^{∗}, Ω_{k}^{∗}, ℐ, d}. For the sake of explicitness, let us note that we are implicitly using the bigrading

C^{k,q}(M) = { u^{α}e_{I} ⊗ dx^{J} : |J| = q, ||I| = k + q }

For p ≧ 2 we shall see that a similar conclusion holds with M replaced by Λ_{Δ}^{p}M where “Λ_{Δ}^{p}” is part of the Δ-multilinear algebra formalism to be discussed in the next section.

2.5.5 3 Δ-Multilinear and Commutative Algebra

The considerations of the preceding section have reduced the computation of the first approximation to the characteristic cohomology to a purely algebraic matter involving a certain type of cohomology constructed from the symbol of the exterior differential system. For p = 1 we find ordinary Spencer cohomology, but for p ≧ 2 a new type of construction appears and it is to this that we now turn, following some preliminary remarks.

3.1 Poincaré polynomials and Spencer cohomology

We keep our previous notation S = ⊕_{k≧0} S^{k}E for the symmetric algebra on an 𝔽-vector space E. We will consider graded S-modules M = ⊕_{k∈ℤ} M^{k} with the properties

(i) M^{k} = 0 for k < k_{0} (ii) dim_{𝔽} M^{k} < ∞ for each k.

Such modules will be called quasi-finitely generated – abbreviated QFG. The modules we shall encounter in practice will generally not be finitely generated. Intuitively we may think of a QFG module as being essentially of the form

M = M_{1} ⊕ M_{2} ⊕ …

where M_{k} is finitely generated in degree d_{k} with d_{1} < d_{2} < …. Many of the usual definitions and properties of finitely generated modules carry over to the QFG case.

For example, for a QFG module M we define the Poincaré series of M by the formula

P_{M}(t) = ∑_{i}(dim_{𝔽} M^{i})t^{i};

it is a Laurent series with a finite tail. If E has dimension n, so that upon choosing a basis e_{1}, …, e_{n} we have

S ≅ 𝔽[e_{1}, …, e_{n}],

then

P_{S}(t) = ∑_{i} ( \begin{matrix}n + i − 1 \\ i \end{matrix} ) t^{i} = \frac{1}{(1 − t)^{n}}

3.1.1 It follows that for any free quasi-finitely generated S-module F

(1 − t)^{n}P_{F} (t) = ∑_{i} f_{i}t^{i}

where f_{i} is the number of free generators of degree i in any set of free generators of F. If we have an exact sequence of QFG S-modules

(1)  0 → M_{ℓ}⟶^{φ_{ℓ}}M_{ℓ−1}⟶^{φ_{ℓ−1}} ⋯ ⟶ M_{1}⟶^{φ_{1}}M_{0} → 0

where φ_{k} has degree δ_{k}, then one has the relation

∑_{k=0}^{ℓ}(−1)^{k}t^{σ_{k}}P_{M_{k}}(t) = 0

where σ_{0} = 0 and σ_{k} = δ_{1} + ⋯ + δ_{k} for k > 0. More generally, if (1) is a complex with homology modules H_{i} = ker φ_{i}/φ_{i+1}(M_{i+1}), then the H_{i} are quasi-finitely generated S-modules and the Euler-Poincaré formula

∑(−1)^{k}t^{σ_{k}}P_{H_{h}}(t) = ∑(−1)^{k}t^{σ_{k}}P_{M_{k}}(t)

is valid.

We want to apply this relation to the Spencer cohomology H^{∗}(M) of a QFG Smodule M. Recall from the preceding section that Spencer cohomology is constructed from the complex of QFG S-modules

⋯ → M ⊗_{𝔽} Λ^{q}E^{∗}⟶^{δ}M ⊗_{𝔽} Λ^{q+1}E^{∗} → ⋯,

and remark that a number of properties from the usual finitely generated case carry over. For example, if S^{+} = ⊕_{k>0} S^{k} denotes the maximal ideal of S then

(2)  H^{n}(M) ≅ M/S^{+}M

which follows immediately from

δ(∑_{i}(−1)^{i−1}m_{i} ⊗ dx^{1} ∧ … ∧ d̂x^{i} ∧ … ∧ dx^{n}) = ∑_{i}(e_{i}m_{i}) ⊗ dx^{1} ∧ … ∧ dx^{n}

3.1.2 More generally, the maximal ideal acts trivially on all of H^{∗}(M), as follows from the homotopy formula

μ_{i} = δν_{i} + ν_{i}δ

where μ_{i} is multiplication by e_{i} and ν_{i}(m ⊗ φ) = m ⊗ e_{i} φ. In particular, the H^{q}(M) are QFG 𝔽-modules – i.e., they are graded 𝔽-vector spaces of the form ⊕_{i≧i_{0}} V_{i} where V_{i} has grade i and dim_{𝔽} V_{i} < ∞.

If F is a QFG free S-module, then it is well known and easy to prove that

(3)  H^{q}(F) = 0,  q < n.

The converse is also true; i.e., if M is QFG and all H^{q}(M) = 0 for q < n then M is free. This may be seen as follows: Setting F = S ⊗_{𝔽} H^{n}(M) we have an exact sequence of QFG modules

0 → N → F → M → 0

with H^{n}(F) →^{∼} H^{n}(M) by construction. It follows from the assumption (3) and long exact cohomology sequence that H^{n}(N) = 0, which by our above observation (2) gives that N = 0 and consequently F ≅ M.

Generalizing this argument gives, as in the usual case, the Hilbert Syzygy theorem for QFG S-modules: Given a QFG module M there is a minimal resolution by free QFG modules

0 → F_{ℓ}⟶^{φ_{ℓ}}F_{ℓ−1}⟶^{φ_{ℓ−1}} ⋯ ⟶ F_{1}⟶^{φ_{1}}F_{0}⟶^{φ_{0}}M → 0,

where φ_{0} has graded degree zero and all the other φ_{i} for i > 0 have positive graded degree, and where

F_{i} ≅ S ⊗_{𝔽} H^{n−i}(M).

We observe that the \underline{resolution length} ℓ satisfies ℓ ≦ n. In particular, for the Poincaré polynomials one has the formula

(1 − t)^{n}P_{M}(t) = ∑_{q}(−1)^{q}t^{q}P_{H^{n−q}(M)}(t)

3.1.3 3.2 Δ-tensor and exterior products

Our basic construction is the Δ-tensor product, denoted M ⊗_{Δ} N, of QFG graded S-modules M and N. The defining properties are

(M ⊗_{Δ} N)^{q} = ⨁_{i+j=q} M^{i} ⊗_{𝔽} N^{j} (i)

e · (m ⊗ n) = em ⊗ n + m ⊗ en, e ∈ E. (ii)

It is easy to see that

e′ · (e · (m ⊗ n)) = e · (e′ · (m ⊗ n))

so that the action of E on M ⊗_{𝔽} N extends to an action of S on M ⊗_{𝔽} N, thereby defining the S-module M ⊗_{Δ} N. The “Leibnitz rule” (ii) is of course what gives interest to this construction. The properties

M ⊗_{Δ} N ≅ N ⊗_{Δ} M
(M ⊕ N) ⊗_{Δ} P ≅ (M ⊗_{Δ} P) ⊕ (N ⊗_{Δ} P)
(M ⊗_{Δ} N) ⊗_{Δ} P ≅ M ⊗_{Δ} (N ⊗_{Δ} P)

hold for the Δ-tensor product of S-modules as for ordinary tensor products.

It is not true that the Δ-tensor product of finitely generated modules is again finitely generated. For example, suppose that S = 𝔽[x]. We will show that

(1)  S ⊗_{Δ} S has {x^{k} ⊗ 1; k ≧ 0} as free generators.

Proof: Let M be the S-submodule generated by the x^{k} ⊗ 1. Then

x · (x^{k} ⊗ x^{ℓ}) − x^{k+1} ⊗ x^{ℓ} = x^{k} ⊗ x^{ℓ+1},

so that x^{k} ⊗ x^{ℓ+1} ∈ M if x^{k+1} ⊗ x^{ℓ} and x^{k} ⊗ x^{ℓ} ∈ M. It follows inductively that M generates S ⊗_{Δ} S. Suppose now that we have a relation

∑_{k} P_{k}(x) · (x^{k} ⊗ 1) = 0

3.2.1 in S ⊗_{Δ} S. Writing P_{k}(x) = ∑_{ℓ} P_{kℓ}x^{ℓ} we look inductively at the terms of total degree zero, one, two, etc. to obtain

P_{00}1 ⊗ 1 = 0 ⟹ P_{00} = 0
P_{01}(x ⊗ 1 + 1 ⊗ x) + P_{10}x ⊗ 1 = 0 ⟹ P_{01} = 0, P_{10} = 0
P_{02}(x^{2} ⊗ 1 + 2x ⊗ x + 1 ⊗ x^{2}) + P_{11}(x^{2} ⊗ x + x ⊗ x) + P_{20}x^{2} ⊗ 1 = 0
⟹ P_{02} = 0, P_{11} = 0, P_{20} = 0

Continuing in this way we find that all P_{k}(x) = 0. Q.E.D. Remark: We may think of (1) as saying that: “By integration by parts in ∫ f^{(k)}(x)g^{(ℓ)}(x)dx we may move all the derivatives to f, and no further reductions are possible”. This is of course obvious. As we shall see below, what is more interesting is what happens when f and g anti-commute.

A generalization of (1) is given by

Proposition 1: For S ≅ 𝔽[e_{1}, …, e_{n}] the Δ-tensor product S ⊗_{Δ} S is a free QFG Smodule, and in fact is freely generated by the 𝔽-subspace S ⊗_{Δ} 1 ⊂ S ⊗_{Δ} S.

Proof: We have the relation of Poincaré polynomials

P_{S⊗_{Δ}S}(t) = P_{S}(t)P_{S}(t) = \frac{1}{(1 − t)^{2n}}

From

e_{i}m ⊗ n ≡ −m ⊗ e_{i}n modulo E · (S ⊗_{Δ} S)

it follows that S ⊗_{Δ} 1 generates the entire S-module S ⊗_{Δ} S. In particular, the S-module mapping

S ⊗_{𝔽} (S ⊗_{Δ} 1) → S ⊗_{Δ} S

given by m⊗(n⊗1) → m·(n⊗1) is surjective. It also preserves natural gradings, and since by the above Poincaré series formula both graded vector spaces have the same dimension in each grade the mapping must be injective as well. Q.E.D.

3.2.2 Corollary: The Δ-tensor product of any number of free QFG S-modules is also free and QFG.

Consider now the q-fold Δ-tensor power of a QFG S-module M, which is denoted by ⊗_{Δ}^{q}M. The symmetric group on q letters, S_{q}, acts on the QFG S-module ⊗_{Δ}^{q}M in the obvious way. Since 𝔽 has characteristic zero, there is a direct sum decomposition of ⊗_{Δ}^{q}M into submodules invariant under the action of S_{q}. These submodules are free in case M is free. One of these summands will be of particular interest to us, namely the q-th Δ-exterior power Λ_{Δ}^{q}M ⊂ ⊗_{Δ}^{q}M.

To get some feeling for this construction suppose that S = 𝔽[x]. We set

m ∧ n = m ⊗ n − n ⊗ m ∈ Λ_{Δ}^{q}S.

Thus by (1) the elements x^{k}∧1 generate the free module Λ_{Δ}^{2}S. But these are not free generators, since we have the relations

x^{0} ∧ 1 = 0  k = 0
x^{2} ∧ 1 = x · (x ∧ 1)  k = 2
x^{4} ∧ 1 = −x^{3} · (x ∧ 1) + 2x · (x^{3} ∧ 1)  k = 4.

We shall see below that this pattern continues, so that we will have the result: Proposition 2: The elements x^{2k+1}∧1, k ≧ 0 freely generate Λ_{Δ}^{2}𝔽[x]. More generally, for S = 𝔽[e_{1}, …, e_{n}] the elements

e_{I} ∧ 1, |I| = 2k + 1

where I = (i_{1}, …, i_{2k+1}) and e_{I} = e_{i_{1}} … e_{i_{2k+1}} freely generate Λ_{Δ}^{2}S. Proof: For any e ∈ E and m, n ∈ S the defining property

e(m ∧ n) = em ∧ n + n ∧ en

implies that (em)∧n ≡ −m∧(en), and so m∧1 ≡ (−1)^{deg m}1∧m where ≡ denotes congruence modulo E · Λ_{Δ}^{2}S. Thus

e_{I} ∧ 1 ≡ (−1)^{|I|+1}e_{I} ∧ 1

3.2.3 and so e_{I} ∧1 ≡ 0 for |I| even. It follows from Proposition 1 that the elements e_{I} ∧1, |I| = 2k + 1, generate Λ_{Δ}^{2}S.

We will first show that these are free generators in the case n = 1 when S ≅ 𝔽[x]. Let V be the graded 𝔽-vector subspace of 𝔽[x] spanned by the x^{2k+1}∧1 for k ≧ 0.

There is a surjective 𝔽[x]-module mapping

𝔽[x] ⊗_{𝔽} V ⟶ Λ_{Δ}^{2}𝔽[x]

given by f ⊗ v → f · v. It will suffice to show that the Poincaré polynomials of these two modules coincide, and inspection shows that each is given by ∑_{ℓ≧0} [\frac{ℓ+1}{2}] t^{ℓ} (in each case, we add one new dimension each time the grade goes up by two).

In general, any relation among the e_{I} ∧1, |I| = 2k + 1, would induce a relation on a line – i.e., when we set e_{i} = α_{i}x where the α_{i} are fixed. If the relation were non-trivial, this would still be the case on a general line in contradiction to the case when S = 𝔽[x] above. Q.E.D.

An alternative argument not reducing to the case n = 1 arises by computing the Poincaré polynomial of Λ_{Δ}^{q}M in terms of the Poincaré polynomial of M, to which we now turn.

Since the Δ-tensor product of two S-modules, when regarded as a vector space, is merely the graded tensor product over 𝔽 of the underlying graded vector spaces, it easily follows that the following standard formula is valid:

Λ_{Δ}^{q}(M ⊕ N) ≃ ⨁_{i+j=q}(Λ_{Δ}^{i}M ⊗_{𝔽} Λ_{Δ}^{j}N).

In particular, if the modules M and N are QFG, this implies the following relation on Poincaré series:

P_{Λ_{Δ}^{q}(M⊕N)}(t) = ∑_{i+j=q} P_{Λ_{Δ}^{i}(M)}(t)P_{Λ_{Δ}^{j}(N)}(t)

The above observation can be used to develop a formula for the Poincaré series of Λ_{Δ}^{q}M in terms of the Poincaré series of M.

First, a few observations about symmetric polynomials will be useful. Let y_{1},y_{2},…,y_{m} be any set of variables and let s_{i} denote the i-th elementary symmetric function of the

3.2.4 variables y_{a}. The power functions of the y_{a} are defined as

p_{i} = (y_{1})^{i} + (y_{2})^{i} + ⋯ + (y_{m})^{i}.

It is well known that the functions p_{i} for i ≤ m freely generate the ring of symmetric functions in the y_{a} and hence that there exist unique polynomial functions E_{i}(p_{1}, p_{2}, …, p_{i}) so that, independent of m, one has

s_{i} = E_{i}(p_{1}, p_{2}, …, p_{i}).

For example, the first four such polynomials are

E_{1} = p_{1}
E_{2} = \frac{1}{2}((p_{1})^{2} − p_{2})
E_{3} = \frac{1}{6}((p_{1})^{3} − 3p_{1}p_{2} + 2p_{3})
E_{4} = \frac{1}{24}((p_{1})^{4} − 6(p_{1})^{2} + 8p_{1}p_{3} + 3(p_{2})^{2} − 6p_{4}).

Now, if y_{0} is another variable, and s′_{i} is the i-th elementary symmetric function of the variables y_{0}, y_{1}, y_{2}, …, y_{m}, then clearly s′_{i} = s_{i} + y_{0}s_{i−1}. It follows that the polynomial functions E_{q} satisfy the following functional identities for any y:

E_{q}(y + p_{1}, y^{2} + p_{2}, …, y^{q} + p_{q}) = E_{q}(p_{1}, p_{2}, …, p_{q}) + yE_{q−1}(p_{1}, p_{2}, …, p_{q−1}).

Proposition 3: For any QFG S-module M, the Poincaré series of its q-th Δ-exterior power Λ_{Δ}^{q}M is given by

P_{Λ_{Δ}^{q}M}(t) = E_{q} (P_{M}(t), P_{M}(t^{2}), …, P_{M}(t^{q})) .

Proof: As has been already remarked, this is really a result about exterior powers of graded vector spaces which are bounded below. Moreover, since the polynomial E_{q} is weighted homogeneous of degree q when the variable p_{i} is assigned the weight i, one can clearly reduce to the case where M^{i} = 0 for all i < 0 by an elementary shift-of-index argument. Thus, we assume this from now on.

3.2.5

Since M^{i} = 0 for all i < 0, it is clear that the i-th graded piece of Λ_{Δ}^{q}M is completely determined by the vector spaces M^{j} for j ≤ i. It follows that it suffices to prove the above formula in the case that M is a finite dimensional graded vector space.

Thus, assume that M is finite dimensional. We will now complete the proof by double induction on q and the dimension of M. If q = 1 or the dimension of M is zero, then the formula is clearly true. If M = N ⊕ 𝔽u where u is an element of degree k, then P_{M}(t) = P_{N}(t) + t^{k}. Moreover, by the above formula for exterior powers of sums, we see that Λ_{Δ}^{q}M ≃ Λ_{Δ}^{q}N ⊕ (𝔽u ⊗ Λ_{Δ}^{q−1}N), so

P_{Λ_{Δ}^{q}M}(t) = P_{Λ_{Δ}^{q}N}(t) + t^{k}P_{Λ_{Δ}^{q−1}N}(t).

This formula, coupled with the functional identity on the polynomials E_{q}, clearly implies the desired result. Q.E.D.

In particular, consider the case of Proposition 2 above when S = 𝔽[e_{1}, …, e_{n}]. Then

P_{Λ_{Δ}^{2}S}(t) = E_{2}(P_{S}(t), P_{S}(t^{2}))

= \frac{1}{2} (\frac{1}{(1 − t)^{2n}} − \frac{1}{(1 − t^{2})^{n}} )

In particular, we have

(1 − t)^{n}P_{Λ_{Δ}^{2}S}(t) = \frac{1}{2} (\frac{1}{(1 − t)^{n}} − \frac{1}{(1 + t)^{n}} )

= ∑_{i∈2ℤ^{+}+1} ( \begin{matrix}n + i − 1 \\ i \end{matrix} ) t^{i}.

The last term is the Poincaré series of any graded 𝔽-subspace of Λ_{Δ}^{2}S which freely generates Λ_{Δ}^{2}S. It follows again that the monomials m∧1 where m has odd degree must be a freely generating set.

Finally, we consider the general free module case where

M ≅ W^{∗} ⊗_{𝔽} 𝔽[e_{1}, …, e_{n}]

for some vector space W^{∗} with basis w^{1}, …, w^{s}. We then have relations generalizing the one given above

(2)  e_{I} w^{α} ∧ w^{β} ≡ (−1)^{|I|+1}e_{I} w^{β} ∧ w^{α},

3.2.6 from which we may infer that

(3) Free generators of Λ_{Δ}^{2}M are given by

e_{I} w^{α} ∧ w^{β} where \{ \begin{matrix}α ≦ β & if |I| is odd \\ α < β & if |I| is even; \end{matrix}

here it is understood that e_{I}w^{α}∧w^{β} = (e_{I}w^{α})∧w^{β}.

In cohomological terms, if S^{+} denotes the maximal ideal of S, then

H^{n}(Λ_{Δ}^{2}M) ≅ Λ_{Δ}^{2}M/S^{+}Λ_{Δ}^{2}M
≅ {φ = φ_{αβ}^{I}e_{I}w^{α} ∧ w^{β}: φ + (−1)^{|I|} ^{t}φ = 0} .

where ^{t}φ = φ_{βα}^{I}e_{I}w^{α}∧w^{β}. A similar result may be proved for the Δ-symmetric product Sym_{Δ}^{2}M, giving that

(4)  H^{n}(Sym_{Δ}^{2}M) ≅ {φ = φ_{αβ}^{I}e_{I} w^{α} ∧ w^{β}: φ − (−1)^{|I|} ^{t}φ = 0} .

We now establish contact between the Δ-exterior product construction and the induced differential on the associated graced to the weight filtration. Namely, comparing formula (6) in Section 2.4 with the formula (1) in that section for the coboundary map in Spencer cohomology using the correspondence given by (6) in Section 2.5, we infer that (7) then extends to the general case, as follows:

(5) We consider the reduction modulo m_{k} of the complex {Ω̄_{k}^{p,∗}, δ}. This is then isomorphic as a complex to the k^{th} graded piece of {C^{∗}(Λ_{Δ}^{p}M) ⊗ T^{∗}, δ}, where M is the graded module associated to the symbol of the exterior differential system.

3.2.7 4. Vanishing of the Characteristic Cohomology

4.1 Characteristic cohomology in the unconstrained case

Recall that the unconstrained case refers to the empty exterior differential system ℐ_{0} on X_{0}; i.e., geometrically we are considering all immersions

f: N → X_{0}

of an n-manifold N into X_{0}. In this case the prolongation tower is described, in the notation of Section 1.3, by

X_{k} = G_{n,k}(X_{0}) = \{ \begin{matrix}all k-jets of immersions of \\ an n-dimensional manifold into X_{0} \end{matrix}\}

We shall also consider the local case; i.e., we work in the inverse image in X = G_{n,∞}(X_{0}) of a particular n-plane E_{0} ∈ G_{n,1}(X_{0}). The basic result here is due, in its general form, to Vinogradov [13]. Theorem. In the unconstrained local case

H̄^{q} = 0 for 0 < q < n.

Proof: We shall see that the result is a direct consequence of the facts, noted in Sections 3.1 and 3.2 above, that

(i) For F a free QFG S-module the Spencer cohomology

H^{q}(F) = 0, q < n.

(ii) If M is a free QFG S-module, then so are the exterior powers Λ_{Δ}^{p}M. Thus by (i)

H^{q}(Λ_{Δ}^{p}M) = 0, q < n.

The first part of the following discussion will apply to the general constrained case. Referring to Section 1.4, we recall that the filtration of Ω^{∗}(X) by the exterior powers of ℐ induces a filtration F^{p} on Ω^{∗}(X) such that the resulting spectral sequence has

E_{1}^{0,q} = H̄^{q}
E_{∞} ⇒ H_{DR}^{∗}(X).

4.1.1 Since we are working in the local case, X is contractible and thus E_{∞}^{p,q} = 0 for p + q > 0. It will therefore suffice to prove that

E_{1}^{p,q} = 0 for p > 0, q < n

(An important point to take note of here is that we are studying the characteristic cohomology \underline{indirectly}, i.e., from the knowledge of the E_{1}^{p,q} for p > 0 and of E_{∞}^{p,q}.)

Set Ω̄^{p,∗} = F^{p}Ω^{∗}/F^{p+1}Ω^{∗} and from Section 2.4 recall the weight filtration defined on Ω̄^{p,∗} for all p but non-trivial only for p > 0. Fixing p > 0, there is the weight spectral sequence Ē_{r}^{k,q} which abuts to E_{1}^{p,q} and has

Ē_{1}^{k,∗} = H(Ω̄{p,∗ \atop k})

where Ω̄{p,∗ \atop k} is the associated graded complex to the weight filtration. Now the main point is that, from (1) in Section 2.4, H(Ω̄{p,∗ \atop k}) is given by sections of a vector bundle over X, and from Section 2.5 we see that

The fiber at ξ ∈ X of H(Ω̄{p,∗ \atop k}) is given by

(1)  H(Ω̄{p,∗ \atop k})_{ξ} = H^{k,∗}(Λ_{Δ}^{p}M)

where M is the graded S-module associated to the symbol of the exterior differential system.

This result is valid for any exterior differential system, and it constitutes the essential step in relating the characteristic cohomology to the symbol of the exterior differential system.

In the unconstrained case the symbol is trivial, and thus as noted above M and the Δ-exterior products are free QFG S-modules. This gives Ē_{1}^{k,q} = 0 for q < n, hence E_{1}^{p,q} = 0 for p > 0, q < n, and from this the desired result. Q.E.D.

Corollary: In the local unconstrained case the sequence

0 → H̄^{n}⟶^{d_{1}}E_{1}^{1,n}⟶^{d_{1}}E_{1}^{2,n}⟶^{d_{1}} ⋯

is exact.

4.1.2

The groups E_{1}^{p,n} are for p > 0 the sections of a vector bundle whose fiber at ξ ∈ X is given by

E_{1,ξ}^{p,n} ≅ Λ_{Δ}^{p}M/S^{+}Λ_{Δ}^{p}M

Thus the fiber has as basis any set of free generators of Λ_{Δ}^{p}M.

Remark: This corollary may be compared with the contact manifold example in Section 1. The (elementary) argument given there dealt with the unprolonged, unconstrained case with one dependent variable (which is equivalent to rank(M) = 1 in the present setting). In that case it identified H̄_{0}^{n} range of a 2^{nd} order linear differential operator. By prolonging, that operator becomes 1^{st} order linear on a suitable jet bundle – it is just the Euler-Lagrange operator d_{1}: H̄^{n} → E_{1}^{1,n} in the present setting. Moreover, the sequence in the corollary gives an exact complex with the Euler-Lagrange operator as the initial term.

The above theorem, together with the corollary, contains completely the classical formalism of Euler-Lagrange operators, Helmholtz operators, and so forth. Moreover, it also contains an “explanation” of the various complicated classical formulae as simply reflecting choices of generators for certain free QFG modules and then expressing other elements in terms of these generators. We shall now explain this. In order to keep the notation as simple as possible, we shall do this in the case of one independent variable x and one dependent variable u.

We thus set u_{0} = u and let

(x, u_{0}, u_{1}, u_{2}, … )

denote the standard coordinates in J^{∞}(ℝ, ℝ), with the contact system having generators

θ = θ_{0} = du_{0} − u_{1}dx
θ_{1} = du_{1} − u_{2}dx
⋮

and structure equations

dθ_{k} = −θ_{k+1} ∧ dx.

4.1.3 In this case M ≅ 𝔽[x] and the statements that we have free generators

(i)  1  for  𝔽[x]
(ii)  {x^{2k+1} ∧ 1}  for  Λ_{Δ}^{2}𝔽[x]
⋮

translate into the statements that the closed differential forms

(o)  dx
(i)  θ_{0} ∧ dx
(ii)  θ_{2k+1} ∧ θ_{0} ∧ dx
⋮

give a framing for H̄^{1}, E_{1}^{1,1}, E_{1}^{2,1}, ….

A class [λ] ∈ H̄^{1} is represented by a 1-form

λ = f(x, u, u_{1}, …, u_{k})dx

and corresponds to a functional

I_{λ}(u) = ∫ f (x, u(x), u′(x), …, u^{(k)}(x)) dx

which, for u with compact support, depends only on the class [λ] in H̄^{1}. As explained in Section 1.4, the Euler-Lagrange equations of this functional are

d_{1}[λ] = 0.

By what we have said above, we will have

[dλ] = [d_{V} λ] = [E(λ) θ_{0} ∧ dx]

for some unique operator E(λ), which is of course the Euler-Lagrange operator. To determine E(λ) we let ≡ denote congruence modulo exact forms, and using the notations and

4.1.4 structure equations from Section 2.1, we have have

d_{1}[λ] = [d_{V} λ]
d_{V} λ = (\frac{∂f}{∂u_{0}} θ_{0} + \frac{∂f}{∂u_{1}} θ_{1} + \frac{∂f}{∂u_{2}} θ_{2} + …) ∧ dx

= \frac{∂f}{∂u_{0}} θ_{0} ∧ dx − \frac{∂f}{∂u_{1}} dθ_{0} − \frac{∂f}{∂u_{2}} dθ_{1} − ⋯

≡ \frac{∂f}{∂u_{0}} θ_{0} ∧ dx − D (\frac{∂f}{∂u_{1}} ) θ_{0} ∧ dx − D (\frac{∂f}{∂u_{2}} ) θ_{1} ∧ dx − ⋯

= (\frac{∂f}{∂u_{0}} − D (\frac{∂f}{∂u_{1}} )) θ_{0} ∧ dx + D (\frac{∂f}{∂u_{2}} ) dθ_{0} + ⋯

≡ (\frac{∂f}{∂u_{0}} − D (\frac{∂f}{∂u_{1}} ) + D^{2} (\frac{∂f}{∂u_{2}} )) θ_{0} ∧ dx − ⋯

⋮

≡ [\frac{∂f}{∂u_{0}} − D (\frac{∂f}{∂u_{1}} ) + D^{2} (\frac{∂f}{∂u_{2}} ) + ⋯ + (−1)^{k}D^{k} (\frac{∂f}{∂u_{k}} )] θ_{0} ∧ dx

where f = f(x, u, u_{1}, …, u_{k}). The operator in the brackets is the classical Euler-Lagrange operator E(λ). The formula form E(λ) is simply a reflection of how one expresses x^{k}, k > 0, in terms of the generator 1 ∈ 𝔽[x].

We note that if k is the largest integer such that \frac{∂f}{∂u_{k}} ≠ 0, then

D^{k} (\frac{∂f}{∂u_{k}} ) = (\frac{∂^{2}f}{∂u_{k}^{2}} ) u_{2k} + (terms depending on (x, u_{0}, u_{1}, …, u_{2k−1})).

Thus, unless f is linear in u_{k}, the Euler-Lagrange operator is of even order 2k.

We now consider a class [E θ_{0}∧dx] ∈ E_{1}^{1,1} where E = E(x, u, u_{1}, …, u_{k}) and ask when E is of the form E(λ) for some λ = f dx. Again this depends only on the class in E_{1}^{1,1} of E θ_{0}∧dx, and by the above corollary the condition is

d_{1}[E θ_{0} ∧ dx] = 0.

By what was said above we will have

[d_{V} (E θ_{0} ∧ dx)] = [∑_{k≧0} H_{2k+1}(E)θ_{2k+1} ∧ θ_{0} ∧ dx]

4.1.5 for unique operators H_{2k+1}(E). The fact that only odd orders appear is a reflection of the fact noted above that in general the Euler-Lagrange operators are of even order. Thus, if E = E(x, u_{0}, u_{1}, …, u_{2k+1}) with ∂E/∂u_{2k+1} ≠ 0, then d_{V} (Eθ_{0}∧dx) ≡ ∂E/∂u_{2k+1}θ_{2k+1}∧θ_{0}∧dx mod{θ_{2k−1}∧θ_{0}, …, θ_{1}∧θ_{0}}, so d_{1}[Eθ_{0}∧dx] ≠ 0 in this case. These operators will express the Helmholtz conditions corresponding to our particular choice of free generators x^{2k+1}∧1 for Λ_{Δ}^{2}𝔽[x], and determining them explicitly will simply reflect how one expresses x^{2k}∧1 in terms of these generators. Explicitly, we have

d_{V} (E θ_{0} ∧ dx) = (\frac{∂E}{∂u_{1}} θ_{1} + \frac{∂E}{∂u_{2}} θ_{2} + \frac{∂E}{∂u_{3}} θ_{3} + \frac{∂E}{∂u_{4}} θ_{4} + ⋯) ∧ θ_{0} ∧ dx
\frac{∂E}{∂u_{2}} θ_{2} ∧ θ_{0} ∧ dx = \frac{∂E}{∂u_{2}} d(θ_{1} ∧ θ_{0}) ≡ −D (\frac{∂E}{∂u_{2}} ) θ_{1} ∧ θ_{0} ∧ dx
\frac{∂E}{∂u_{4}} θ_{4} ∧ θ_{0} ∧ dx = \frac{∂E}{∂u_{4}} d(θ_{3} ∧ θ_{0}) − \frac{∂E}{∂u_{4}} θ_{3} ∧ θ_{1} ∧ dx

≡ −D (\frac{∂E}{∂u_{4}} ) θ_{3} ∧ θ_{0} ∧ dx − (\frac{∂E}{∂u_{4}} ) d(θ_{2} ∧ θ_{1})

≡ −D (\frac{∂E}{∂u_{4}} ) θ_{3} ∧ θ_{0} ∧ dx + D (\frac{∂E}{∂u_{4}} ) θ_{2} ∧ θ_{1} ∧ dx

≡ −2D (\frac{∂E}{∂u_{4}} ) θ_{3} ∧ θ_{0} ∧ dx + D (\frac{∂E}{∂u_{4}} ) d(θ_{2} ∧ θ_{0})

≡ −2D (\frac{∂E}{∂u_{4}} ) θ_{3} ∧ θ_{0} ∧ dx − D^{2} (\frac{∂E}{∂u_{4}} ) θ_{2} ∧ θ_{0} ∧ dx

≡ −2D (\frac{∂E}{∂u_{4}} ) θ_{3} ∧ θ_{0} ∧ dx − D^{2} (\frac{∂E}{∂u_{4}} ) d(θ_{1} ∧ θ_{0})

≡ −2D (\frac{∂E}{∂u_{4}} ) θ_{3} ∧ θ_{0} ∧ dx + D^{3} (\frac{∂E}{∂u_{4}} ) θ_{1} ∧ θ_{0} ∧ dx

⋮

In general, if we write

x^{2k} ∧ 1 = ∑_{ℓ≧0} C_{ℓ}^{k}x^{2k−2ℓ−1} · (x^{2ℓ+1} ∧ 1)

then

H_{2ℓ+1}(E) = \frac{∂E}{∂u_{2ℓ+1}} − ∑_{k} C_{ℓ}^{k}D^{2k−2ℓ−1} (\frac{∂E}{∂u_{2k}} )

4.1.6 This shows, in the most direct manner, how the Δ-exterior algebra results directly translate into classical formulae, in this case, the Helmholtz conditions H_{2ℓ+1}(E) = 0 which are the necessary and sufficient conditions that an expression E be of the form E = E(λ) for some functional I_{λ}. A different choice of generators for the module Λ_{Δ}^{2}𝔽[x] would yield a different, albeit equivalent, set of equations.

4.1.7 4.2 Involutivity and Δ-exterior powers

Let M be a QFG graded S-module with minimal resolution

(1)  0 → F_{ℓ}⟶^{φ_{ℓ}}F_{ℓ−1}⟶^{φ_{ℓ−1}} → ⋯ → F_{1}⟶^{φ_{1}}F_{0}⟶^{φ_{0}}M → 0

by QFG free graded modules F_{ν}. Recall that ℓ ≦ n and that

φ_{0}: F_{0}^{k} → M^{k}
φ_{ν}: F_{ν}^{k} → ⊕_{ℓ≧k+1} F_{ν−1}^{ℓ},  ν ≧ 1.

The first of these simply states that φ_{0} maps free generators of F_{0} onto a set of generators of M, while the second says that we have chosen a minimal set of free generators of F_{0} with the property, that we then choose a minimal set of generators for the relations among the generators of M, and so forth.

Definition: (i) M is involutive if each φ_{ν}, ν ≧ 1, has graded degree exactly equal to one. (ii) the integer ℓ giving the length of the resolution (1) will be called the characteristic

number of the module M.

Intuitively, M is involutive if the relations among a minimal set of generators of M are themselves minimally generated by relations in degree one, if the relations among the relations are themselves minimally generated in degree one, and so forth. If M is generated in degree zero, then involutivity is equivalent to

H^{+,q}(M) = 0,

where H^{q}(M) = ⊕_{k} H^{k,q}(M) is the graded vector space of Spencer cohomology. Whether or not M is generated in degree zero, the characteristic number ℓ(M) is defined by

ℓ(M) = min{ℓ : H^{q}(M) = 0 for 0 ≦ q < n − ℓ}

Below we shall recall the definition of the Cartan characters s_{0}, s_{1}, …, s_{n} of the first prolongation of an exterior differential system. These integers satisfy

s_{0} ≧ s_{1} ≧ … ≧ s_{n}

4.2.1 and are easily computed in practice. Setting

σ_{i} = s_{i} − s_{i+1}, i = 0, 1, …, n − 1

and σ_{n} = s_{n}, the characteristic number ℓ ≥ 0 is also uniquely characterized as the smallest integer so that

(2)  σ_{n−ℓ} ≠ 0,

and this is how it is computed in geometric examples. We remark that although the Cartan characters are not invariant under prolongation, the characteristic number defined by (2) is invariant.

Our main algebraic result is given by the

Theorem 1: If M is involutive, then so are the Δ-exterior powers Λ_{Δ}^{p}M. Moreover,

(3)  ℓ(Λ_{Δ}^{p}M) ≦ ℓ(M)

Actually, as our proof will show, the inequality ℓ(Λ_{Δ}^{p}M) ≦ ℓ(M) holds without the assumption of involutivity and, once formulated and set up, this result by itself is relatively easy.

More substantial is to understand the minimal resolution of Λ_{Δ}^{p}M in terms of that of M. As we will see below, if there are “sufficiently many” variables and if the symbol is generic among involutive symbols, then Λ_{Δ}^{2}M will be freely generated in low degrees but will have relations in higher degrees. It is perhaps for this reason that our proof of the theorem stated above is not intrinsic.

A geometric application results from the fact that

H^{q}(Λ_{Δ}^{p}M) = 0, 0 ≦ q < n − ℓ

Before proving Theorem 1, we shall make a few preliminary remarks. For these we will let ℐ_{0} be an exterior differential system on a manifold X_{0} and E_{0} ⊂ T_{p}X_{0} an n-dimensional integral element of ℐ_{0}. The concept of what it means for E_{0} to be ordinary may be defined (see pages 73–ff in [2]), and ℐ_{0} will be said to be involutive in a neighborhood of an

4.2.2 ordinary integral element (if such exists). If E_{0} is ordinary, then the symbol is involutive in a neighborhood U of E_{0} in G_{n}(ℐ_{0}), and moreover the prolongation tower will satisfy the regularity assumption of Section 1.3 over U. Here, it will be convenient to set X_{0} = U and begin the construction of the prolongation tower with X_{1}, and when this is done we will say that we are in the local, involutive case.

Finally, we remark that in practice there is a very effective criterion called Cartan’s test that allows us to determine when an integral element E_{0} is ordinary in terms of the Cartan characters s_{1}, s_{2}, …, s_{n} mentioned above. We shall review this at an appropriate time. The upshot is that in practice both the issue of involutivity and the characteristic number may be effectively determined from the characters s_{i}.

The following application of the theorem follows immediately from the same argument as in the preceding section.

Theorem: In the local, involutive case the characteristic cohomology satisfies

H̄^{q} = 0 for 0 < q < n − ℓ

where ℓ is the characteristic number of the symbol of the exterior differential system.

In the context of fibered manifolds what essentially amounts to the case ℓ = 1 of this result is due to Vinogradov [13]; he refers to it as the two-line theorem. An alternative proof of Vinogradov’s result is proposed in [11], but we are unable to follow the argument. Generalizations of Vinogradov’s theorem are also given in [1] and [12]. All of these works deal with the situation of fibered manifolds and as so formulated do not imply the above result.

In light of the above result, attention is naturally focussed on the first possibly nonvanishing cohomology groups.

Definition: In the local, involutive case we set 𝒞 = H̄^{n−ℓ} and shall call 𝒞 the group of conservation laws of the exterior differential system.

Much of our subsequent work will be devoted to studying the group of conservation laws in classes of examples and to geometric applications of those results. From the

4.2.3 characteristic spectral sequence we have

(4)  𝒞 ≅ ker d_{1}: E_{1}^{1,n−ℓ} → E_{1}^{2,n−ℓ}

This rather abstract looking result will in practice turn out to be extremely important. In effect, this will allow us to systematically study the conservation laws in terms of the symbol, then the sub-principal symbol and so forth. Moreover, when written out, (4) will give the conservation laws as the kernel of a certain canonical linear differential operator and not as a quotient space – one may think in effect of having an intrinsic harmonic operator for ordinary deRham cohomology. This puts in a general context the phenomena noted in the examples in Section 1.1.

4.2.4 4.3 The tableau and its normal form

We will begin the proof of Theorem 1 in the preceding section. As mentioned there, our proof is not intrinsic but will give more than just the statement of the theorem. However, we can explain why the estimate ℓ(Λ_{Δ}^{p}(M) ≦ ℓ(M) should be true, as follows: “The characteristic number of M is the smallest integer ℓ such that if we restrict to a generic n−ℓ plane then M is free. By Proposition 2 in Section 3.2, the restriction of Λ_{Δ}^{p}M to this n − ℓ plane will also be free, hence the result”. This reasoning is not correct, but it is also not entirely incorrect and captures the essence of the result.

We now recall the device known as a “tableau” introduced by Cartan and used by him to put a symbol in a sort of normal form – further details may be found in pages 141 ff in [2]. Let W and E be vector spaces, B ⊂ W^{∗} ⊗ E a subspace, and B. ⊂ W^{∗} ⊗ SE the submodule generated by B. Then B is a symbol and M = W^{∗} ⊗ SE/B. is the graded S-module associated to B.

We may think of B as corresponding to a linear, homogeneous constant coefficient PDE system whose formal power series solutions are given by the graded subspace

A· ⊂ W ⊗ SE^{∗}

annihilated by B. Thus A· = ⊕_{k≧0} A^{k} where A^{k} ⊂ W ⊗S^{k}E^{∗} are the W -valued homogeneous polynomials P of degree k that satisfy

B · P = 0

where B · P ∈ S^{k−1}E^{∗}. We shall call A = A^{1} the tableau associated to B and shall think of elements L ∈ A as W -valued linear functions on the vector space E. Choosing a generic linear coordinate system x^{1}, …, x^{n} we may then ask how many elements L ∈ A are uniquely determined by their restrictions to linear subspaces x^{k+1} = ⋯ = x^{n} = 0 and whether the restriction of L to this subspace may be arbitrarily prescribed.

To visualize this we let w^{1}, …, w^{s} ∈ W^{∗} be a basis, to be adjusted during the following construction, and e_{1}, …, e_{n} ∈ E the basis dual to the generic linear coordinate system x^{1}, …, x^{n}, which is fixed once and for all. We denote by

b^{λ} = b_{α}^{λi}w^{α} ⊗ e_{i} ∈ W^{∗} ⊗ E

4.3.1 a basis for B and introduce the spanning elements θ_{i}^{α} for the dual space A^{∗} given by projecting w^{α} ⊗ e_{i} to M^{1} = A^{∗}. Then the θ_{i}^{α} are subject to the defining relations

b_{α}^{λi}θ_{i}^{α} = 0.

We then introduce the tableau matrix

θ = ∥ \begin{matrix}θ_{1}^{1} & θ_{2}^{1} & ⋯ & θ_{n}^{1} \\ ⋮ & ⋮ & & ⋮ \\ θ_{1}^{s} & θ_{1}^{s} & ⋯ & θ_{n}^{s} \end{matrix} ∥

By definition, the Cartan characters s_{0}, s_{1}, s_{2}, …, s_{n} are given by s_{0} = s and

s_{1} + ⋯ + s_{k} = \{ \begin{matrix}number of independent 1-forms \\ θ_{i}^{α} in the first k columns of θ \end{matrix}\}

We first choose our basis w^{α} for W^{∗} so that θ_{1}^{1}, …, θ_{1}^{s_{1}} are linearly independent, we then refine that choice so that θ_{1}^{1}, …, θ_{1}^{s_{1}}, θ_{2}^{1}, …, θ_{2}^{s_{2}} are linearly independent, and so forth (see page 141 of [2] for further discussion). Then s_{1} ≧ s_{2} ≧ ⋯ ≧ s_{n} and s_{1} + ⋯ + s_{n} = dim A. We may picture the normalized tableau matrix as looking like

∥ \begin{matrix}θ_{1}^{1} & θ_{2}^{1} & & θ_{n}^{1} \\ ⋮ & ⋮ & ⋯ & θ_{n}^{s_{n}} \\ & & & ∗ \\ & θ_{2}^{s_{2}} & & ∗ \\ θ_{1}^{s_{1}} & ∗ & & ∗ \\ ∗ & ∗ & & ∗ \end{matrix} ∥

where for β > s_{k} the form θ_{k}^{β} is a linear combination of the other θ_{i}^{α} where i ≦ k and α ≦ s_{i}, i.e., the ∗ forms are linear combinations of the other θ’s.

Actually, for the purposes of the argument we shall be giving, it will be advantageous to organize the normalized tableau matrix into blocks

∥  ∥

4.3.2 where the s_{i} are constant in each block. We will develop the proof of our result through examination of a number of special cases. Before turning to this we will introduce some notations.

• dx^{i} and θ_{I}^{α} are linear functions on two vector spaces, the dx^{i} being linearly independent and the θ_{I}^{α} subject to the relations

b_{α}^{λi}θ_{Ii}^{α} = 0

• Ω^{p,q} is spanned by the forms

θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ dx^{j_{1}} ∧ … ∧ dx^{j_{q}}

and Ω^{∗} = ⊕Ω^{p,q}

• The differential

d: Ω^{p,q} → Ω^{p,q+1}

is defined by d(dx^{i}) = 0 and

dθ_{I}^{α} = −θ_{Ii}^{α} ∧ dx^{i},

together with the usual properties of being a derivation of the exterior algebra

• We will denote by Λ_{m}^{∗} = ⊕Λ_{m}^{p,q} a complex analogous to Ω^{p,q} but where the number of dx^{i}’s is m and where there are no relations on the θ_{I}^{α} – this “free” case models the unconstrained geometric situation in which the number of independent variables is m, and we have seen that

(1)  H^{k}(Λ_{m}^{∗}) = 0,  k < m

4.3.3 4.4 The determined case

By definition, the determined case is an exterior differential system whose Cartan characters are the same as for a determined PDE system

∂u/∂x^{n} = \{ \begin{matrix}constant linear combination \\ of ∂u/∂x^{1}, …, ∂u/∂x^{n−1} \end{matrix}

where u = ^{t}(u^{1}, …, u^{s}). The Cartan characters are given by

s_{1} = ⋯ = s_{n−1} = s, s_{n} = 0.

In this case we may solve for the last column \underline{θ}_{n} of the tableau matrix in terms of the first n − 1 columns \underline{θ}_{ρ}, 1 ≦ ρ ≦ n − 1. Thus we have relations

\underline{θ}_{n} = b^{ρ}\underline{θ}_{ρ}

where each b^{ρ} is an s × s matrix. These relations induce relations

\underline{θ}_{In} = b^{ρ}\underline{θ}_{I_{ρ}}

on the vector-valued forms \underline{θ}_{J} = ^{t}(…, θ_{J}^{α}, …). When written out fully, these relations are

θ_{n}^{α} = b_{β}^{αρ}θ_{ρ}^{β}
θ_{In}^{α} = b_{β}^{αρ}θ_{Iρ}^{β}

We shall refer to the case b^{ρ} = 0 as the split case; on the symbol level it looks like the PDE system ∂u/∂x^{n} = 0. The opposite extreme is when the b^{ρ} are generic.

It is well known and easy to prove that any determined exterior differential system is involutive.

The minimal free resolution of the graded S-module associated to the symbol is

0 → W_{1,.}^{∗}⟶^{b}W.^{∗} → M → 0

where W^{∗}, W_{1}^{∗} are s-dimensional vector spaces with respective bases w^{α}, w_{1}^{α} and where

b(w_{1}^{α}) = e_{n}w^{α} − b_{β}^{αρ}e_{ρ} · w^{β}

4.4.1 corresponding to (1). We want to infer properties of the minimal free resolution of the Λ_{Δ}^{p}M_{s}. For this we will use the standard shift notation P [ℓ] for a graded object P – thus P = ⊕_{k} P^{k} and P [ℓ] = ⊕_{k} P [ℓ]^{k} where P [ℓ]^{k} = P^{k+ℓ}. We introduce the complexes

Ω′^{p,q} = dx^{n} ∧ Ω′^{p,q−1} ≅ Λ_{n−1}^{p,q−1}[1]
Ω′′^{p,q} = Ω^{p,q}/Ω′^{p,q}  ≅ Λ_{n−1}^{p,q}

The first line means that Ω′^{∗} is a subcomplex of Ω^{∗} which is isomorphic as a complex to Λ_{n−1}^{∗}[1], this being the case since there are no relations among the θ_{ρ_{1}...ρ_{k}}^{α}. In the second line, Ω′′^{∗} is in effect obtained by setting dx^{n} = 0, and it is again isomorphic as a complex to Λ_{n−1}^{∗}. From the exact cohomology sequence of

0 → Ω′^{∗} → Ω^{∗} → Ω′′^{∗} → 0

together with (1) in the preceding section, we obtain

H^{p,q}(Ω^{∗}) = 0, q ≦ n − 2

(1) \begin{aligned}0 & → & H^{p,n−1}(Ω^{∗}) & →H^{p,n−1}(Ω′′^{∗}) & ⟶^{δ} & H^{p,n}(Ω′^{∗}) & → H^{p,n}(Ω^{∗}) & → & 0 \\ & & & ∥_{≀} & & ∥_{≀} \\ & & & H^{p,n−1}(Λ_{n−1}^{∗}) & & H^{p,n−1}(Λ_{n−1}^{∗}[1]) \end{aligned}

The first result implies (3) in Theorem 1 in Section 4.2, which as one sees is relatively elementary once one sets the problem up and has the statement that the Λ_{Δ}^{p}M are free in case M is.

The exact sequence (1) will now be examined, and we shall see a great difference between the cases p = 1 and p ≧ 2. In the former we will see that δ = 0, just as if we were in the split case; in the latter almost the opposite will hold.

We first compute δ in the case p = 1. Setting dx_{i} = (−1)^{i−1}dx^{1}∧ … ∧d̂x^{i}∧ … ∧dx^{n}, dx′ = dx_{n}, dx′_{ρ} = (−1)^{ρ}dx^{1}∧ … ∧d̂x^{ρ}∧ … ∧dx^{n−1} and dx = dx^{1}∧ … ∧dx^{n} we have bases

θ^{α} ∧ dx′  for  H^{1,n−1}(Ω′′^{∗})
θ^{α} ∧ dx  for  H^{1,n}(Ω′^{∗})  (dx = dx^{n} ∧ dx′)

4.4.2 Now

δ(θ^{α} ∧ dx′) = class of d(θ^{α} ∧ dx′)
= class of −θ_{n}^{α} ∧ dx
= class of d(b_{β}^{αρ}θ^{β} ∧ dx_{ρ}), 1 ≦ ρ ≦ n − 1,
= class of (−1)^{n−1}dx^{n} ∧ d(b_{β}^{αρ}θ^{β} ∧ dx′_{ρ})
= 0.

Thus when p = 1 there is no difference between the split and the general case.

We next turn to the case p = 2. Let E′ be the subspace dx^{n} = 0 of E with basis e_{1}, …, e_{n−1}, S′ = SE′ the symmetric algebra of E′, and M′ = W′.^{∗} = W^{∗} ⊗_{𝔽} S′. Then as we have seen at the end of Section 3.2, Λ_{Δ}^{2}M′ has generators

e_{I}w^{α} ∧ w^{β}

where I = (ρ_{1}, …, ρ_{k}) and relations

e_{I}w^{α} ∧ w^{β} + (−1)^{k}e_{I}w^{β} ∧ w^{α} ≡ 0

among the generators, where ≡ is congruence modulo the action of the maximal ideal S′^{+} of SE′. Thus we shall write elements of V′ =: Λ_{Δ}^{2}M′/S^{+} · Λ_{Δ}^{2}M′ as

(2)  φ = φ_{αβ}^{I}e_{I}w^{α} ∧ w^{β}, φ + (−1)^{|I|}^{t}φ = 0.

where (^{t}φ)_{αβ}^{I} = φ_{βα}^{I}. Now each of H^{2,n−1}(Ω′′^{∗}) and H^{2,n}(Ω′^{∗}) is isomorphic to V′, and making this identification we shall compute δ: V′ → V′. For this we introduce indeterminates ξ_{1}, …, ξ_{n−1} and set

φ(ξ) = ∥φ_{αβ}(ξ)∥
= ∥φ_{αβ}^{I}ξ_{I}∥
b(ξ) = ∥b_{β}^{αρ}ξ_{ρ}∥

Proposition: With the above notations

(δφ)(ξ) = [φ(ξ), b(ξ)] ( = ^{t}b(ξ)φ(ξ) − φ(ξ)b(ξ))

4.4.3 Proof: We set

\underline{φ} = φ_{αβ}^{I}θ_{I}^{α} ∧ θ^{β} ∧ dx′ ∈ H^{2,n−1}(Ω′′^{∗}) ≅ V′

and must compute

δ(\underline{φ}) = class of dφ ∈ H^{2,n}(Ω′^{∗}) ≅ V′.

Now we have (note the signs)

d(θ_{I}^{α} ∧ θ^{β} ∧ dx′) = θ_{In}^{α} ∧ θ^{β} ∧ dx + θ_{I}^{α} ∧ θ_{n}^{β} ∧ dx
= (b_{γ}^{αρ}θ_{Iρ}^{γ} ∧ θ^{β} + b_{γ}^{βρ}θ_{I}^{α} ∧ θ_{ρ}^{γ}) ∧ dx
≡ (b_{γ}^{αρ}θ_{Iρ}^{γ} ∧ θ^{β} − b_{γ}^{βρ}θ_{Iρ}^{α} ∧ θ^{γ}) ∧ dx

where ≡ is congruence modulo exact forms and we have used

d(θ_{I}^{α} ∧ θ^{γ} ∧ dx_{ρ}) = (θ_{Iρ}^{α} ∧ θ^{γ} + θ_{I}^{α} ∧ θ_{ρ}^{γ}) ∧ dx

The proposition follows by using the symmetry conditions (2) on φ. Q.E.D.

Discussion: M has a two-step minimal free resolution with minimal generators and minimal generating relations given respectively by

w^{α} (i) e_{n}w^{α} − b_{β}^{αρ}e_{ρ} · w^{β} = 0 (ii)

Using the proposition, we will show that Λ_{Δ}^{2}M has a minimal free resolution

(3)  0 → F_{1}⟶^{λ}F_{0} → Λ_{Δ}^{2}M → 0

where F_{1} ≠ 0 and λ has degree one. It follows that:

Λ_{Δ}^{2} is involutive and ℓ(Λ_{Δ}^{2}M) = ℓ(M).

We will also see that in the split case F_{1} has generators in all degrees where F_{0} does, but in the generic case generators of F_{1} appear only in degrees that are high relative to s and n; i.e., in low degrees the generators of Λ_{Δ}^{2}M generate free summands.

4.4.4

In order to best understand the situation, we will work through the first few degrees, and from these draw general conclusions based on the evident patterns that will emerge.

Degree zero: Minimal generators and relations among the generators are

w^{α} ∧ w^{β}, α < β (i)_{0} e_{n}(φ_{αβ}w^{α} ∧ w^{β}) − e_{ρ} · (φ_{αγ}b_{β}^{γρ}w^{α} ∧ w^{β}) = 0 where (ii)_{0} φ +^{t}φ = 0, [φ, b^{ρ}] = 0 for ρ = 1, …, n − 1 (ii)′_{0}

Thus, when we try to “lift” the relations (ii) from M to Λ_{Δ}^{2}M we are only able to lift certain linear combinations (ii)_{0}, namely those which satisfy the equations (ii)′_{0}.

Degree one: Minimal generators are a linearly independent set of elements

(i)_{1}  ψ = ψ_{αβ}^{ρ}e_{ρ} · w^{α} ∧ w^{β},  ψ −^{t}ψ = 0

taken modulo those of the form

(i)′_{1}  ψ^{ρ} = [φ, b^{ρ}],  φ +^{t}φ = 0

Relations among these generators are given by

(ii)_{1}  e_{n} · (ψ_{αβ}^{ρ}e_{ρ} · w^{α} ∧ w^{β}) − e_{σ} · (ψ_{αγ}^{ρ}b_{β}^{γσ}e_{ρ} · w^{α} ∧ w^{β}) = 0

where

(ii)′_{1}  [b^{σ}, ψ^{ρ}]e_{ρ}e_{σ} = 0.

Thus we find that the “expected” generators (i)_{1} must be corrected by “moding out” by the range (i)′_{1} of the operator φ → [φ, b^{ρ}] whose kernel gave (ii)′_{0}, and moreover that trying to lift the relations (ii) as in (ii)_{1} requires that the condition (ii)′_{1} be satisfied.

Degree two: Minimal generators are a linearly independent set of elements

(i)_{2}  ψ = ψ_{αβ}^{ρσ}e_{ρ}e_{σ}w^{α} ∧ w^{β},  ψ +^{t}ψ = 0

4.4.5 taken modulo those of the form

(i)′_{2}  ψ^{ρσ} = [b^{ρ}, φ^{σ}] + [b^{ρ}, φ^{σ}],  φ −^{t}φ = 0.

Relations among these generators are

(ii)_{2}  e_{n} · (ψ_{αβ}^{ρσ}e_{ρ}e_{σ}w^{α} ∧ w^{β}) − e_{τ} · (ψ_{αγ}^{ρσ}b_{β}^{γτ} e_{ρ}e_{σ}w^{α} ∧ w^{β}) = 0

where

(ii)′_{2}  [b^{ρ}, ψ^{στ}]e_{ρ}e_{σ}e_{τ} = 0

From these first few terms the general pattern is clear. The statements (i)_{i}, (i)′_{i}, (ii)_{i}, (ii)′_{i} are interpretations of the exact cohomology sequence with the various identifications being made. As far as the involutivity goes, from (1) it follows that for all p there is a two-step minimal free resolution

0 → F_{1,p} → F_{0,p} → Λ_{Δ}^{p}M → 0

where as graded vector spaces

F_{0,p}/S^{+}F_{0,p} ≅ coker δ
F_{1,p}/S^{+}F_{1,p} ≅ ker δ
deg δ = 1

The mapping δ probes when a linear combination

φ = φ_{α_{1}...α_{p}}^{I_{1}...I_{p−1}}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p−1}}^{α_{p−1}} ∧ θ^{α_{p}} ∧ dx_{n}, |I_{1}| + ⋯ + |I_{p−1}| + 1 = k

of elements of H^{p,n−1}(Λ_{n−1}^{∗}) can be lifted to a closed form

ψ = φ + η_{α_{1}...α_{p}}^{I_{1}...I_{p−1,ρ}}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p−1}}^{α_{p−1}} ∧ θ^{α_{p}} ∧ dx_{ρ}, |I_{1}| + ⋯ + |I_{p−1}| + 1 = k

of the same weight k − n + 1 as φ. Writing

0 = dψ = E dx

4.4.6 we see that the formula φ → E has degree one, i.e., the relations among the generators of Λ_{Δ}^{p}M are themselves generated in degree one.

Finally, again in the case p = 2 we will make some observations about the operator

(3)  φ(ξ) → [φ(ξ), b(ξ)]

If, say, degree φ(ξ) = 2k−1 is odd, then this is an operator from a vector space of dimension ( \begin{matrix}n−2+2k \\ 2k−1 \end{matrix} ) \frac{s(s+1)}{2} to a vector space of dimension ( \begin{matrix}n−1+2k \\ 2k \end{matrix} ) \frac{s(s−1)}{2}. The inequality

( \begin{matrix}n − 1 + 2k \\ 2k \end{matrix} ) \frac{s(s − 1)}{2} ≧ ( \begin{matrix}n − 2 + 2k \\ 2k − 1 \end{matrix} ) \frac{s(s + 1)}{2}

is for s ≧ 2 equivalent to

1 + \frac{n − 1}{2k} ≧ 1 + \frac{2}{s − 1}

For fixed s and n this inequality is satisfied for

k ≦ (s − 1)(n − 1)/4

but not for k > (s − 1)(n − 1)/4. Thus we may expect that for a generic symbol the operator (4) will be injective for small k but not for large k. This means that in the minimal free resolution (3) the minimal generators of Λ_{Δ}^{2}M in low degree will be free – i.e., Λ_{Δ}^{2}M will have free summands generated in the lower degrees – but that there will definitely be degree one relations among the high degree minimal generators of Λ_{Δ}^{2}M. In conclusion, we may see that Λ_{Δ}^{2}M is involutive and in general will be a mix of free modules generated in low degree together with involutive modules of resolution length one generated in all higher degrees.

4.4.7 4.5 The unmixed, overdetermined case

The definition this is an involutive exterior differential system whose Cartan characters are the same as for an overdetermined PDE system which is not a non-trivial iteration of several systems; it is expressed by the condition

(1)  s_{1} = ⋯ = s_{n−ℓ} = s, s_{n−ℓ+1} = … = s_{n} = 0,

which is equivalent to the tableau having the block form

∥ \begin{matrix}∗ ⋯ ∗ & 0 ⋯ 0 \\ ⋮  ⋮ & ⋮  ⋮ \\ ∗ ⋯ ∗ & 0 ⋯ 0 \\ \overbrace{n − ℓ} & \overbrace{ℓ} \end{matrix} ∥ .

In this case we may solve for the last ℓ columns of the tableau matrix in terms of the first n − ℓ, expressed by the relations

\underline{θθθ}_{λ} = b_{λ}^{ρ}\underline{θθθ}_{ρ} 1 ≦ ρ, σ ≦ n − ℓ, n − ℓ + 1 ≦ λ, μ ≦ n

Written out fully these relations are

(2)  θ_{λ}^{α} = b_{λβ}^{ρα}θ_{ρ}^{β}

Setting B_{λ}(ξ) = b_{λ}^{ρ}ξ_{ρ}, the involutivity of the symbol is equivalent to the commutation relations

(3)  [B_{λ}(ξ), B_{μ}(ξ)] = 0

The minimal free resolution of the graded S-module associated to the symbol is

0 → W_{ℓ,.}^{∗}⟶^{φ_{ℓ}} → ⋯ → W_{1,.}^{∗}⟶^{φ_{1}}W^{∗} → M → 0.

As will now be explained, this is essentially the Koszul complex constructed from the commuting “variables” B_{λ}(ξ). For this we introduce a vector space V of dimension ℓ and having a basis v_{λ} where n − ℓ + 1 ≦ λ ≦ n. We set

W_{k}^{∗} = W^{∗} ⊗_{𝔽} Λ^{k}V

4.5.1 and define the mapping

W_{1,.}^{∗}⟶^{φ_{1}}W.^{∗}

to be that induced by the assignment of generators

(4)  φ_{1}(w^{α} ⊗ v_{λ}) = e_{λ}w^{α} − b_{λβ}^{ρα}e_{ρ}w^{β}

based on (2). Due to the commutation relations (3), this mapping induces

W^{∗} ⊗ SV → W.^{∗}

and allows us to define a degree one mapping

W.^{∗} ⊗ Λ^{k}V → W.^{∗} ⊗ Λ^{k−1}V

given by

e_{I}w^{α} ⊗ v_{λ_{1}} ∧ … ∧ v_{λ_{k}} → ∑_{ν}(−1)^{ν−1}e_{I}φ_{1}(w^{α} ⊗ v_{λ_{ν}})v_{λ_{1}} ∧ … ∧ v̂_{λ_{ν}} ∧ … ∧ v_{λ_{k}}

where φ_{1} is (4). This defines the above complex, which can be seen to be exact. We will now show that

(5)  H^{q}(Λ_{Δ}^{p}M) = 0,  0 ≦ q < n − ℓ

For this we retain the notations used in Section 3.6, and we filter the complex Ω^{∗} as follows: Set

\{ \begin{matrix}y^{ρ} & = x^{ρ} & 1 ≦ ρ, σ ≦ n − ℓ \\ t^{λ} & = x^{λ} & n − ℓ + 1 ≦ λ, μ ≦ n, \end{matrix}

and write a general φ ∈ Ω^{∗} as

φ = φ_{α_{1}...α_{p}RK}^{I_{1}...I_{p}}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ dy^{R} ∧ dt^{K}, |I_{ν}| ⊂ {1, …, n − ℓ},

and then define

¯F̄^{a}Ω^{∗} = {φ: |K| ≧ a}

4.5.2 That is, ¯ F̄^{a}Ω^{∗} consists of forms that have at least a differentials selected from the last n − ℓ coordinates. Clearly

¯F̄^{a+1}Ω^{∗} ⊂ ¯ F̄^{a}Ω^{∗}
d( ¯F̄^{a}Ω^{∗}) ⊂ ¯ F̄^{a}Ω^{∗}

while the symbol relations (2) imply that

(6)  ¯  F̄^{a}Ω^{∗}/ ¯F̄^{a+1}Ω^{∗} ≅ ⊕Λ_{n−ℓ}^{∗−a}

More precisely, we identify Λ_{n−ℓ}^{∗} with the complex of forms

ψ = ψ_{α_{1}...α_{p}R}^{I_{1}...I_{p}}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ dy^{R}, I_{ν} ⊂ {1, …, n − ℓ}.

Then for each multi-index K with |K| = a, we have that dt^{K} ∧ψ ∈ ¯ F̄^{a}Ω^{∗} and this induces the isomorphism of vector spaces

¯F̄^{a}Ω^{∗}/ ¯F̄^{a+1}Ω^{∗} ≅ ⊕_{|K|=a} dt^{K} ∧ Λ_{n−ℓ}^{∗−a}

inducing the isomorphism of complexes (6). The spectral sequence of the above filtered complex has

¯Ē{a,b \atop ∞} ⟹ H^{a+b}(Ω^{∗})
¯Ē{a,b \atop 1} = 0 for 0 ≦ b < n − ℓ + a

and this implies (5). Remark: The proof shows that the vanishing result (5) does not require the involutivity of the symbol; it holds whenever the Cartan characters satisfy (1). This will be a completely general phenomenon – the inequality ℓ(Λ_{Δ}^{p}M) ≦ ℓ(M) does not require that the symbol be involutive. The much more subtle question of relating the minimal resolution of Λ_{Δ}^{p}M to that of M will now be discussed in the first non-trivial overdetermined case ℓ = 2.

For this we write the relations (2) as

θ_{n}^{α} = b_{β}^{αρ}θ_{ρ}^{β}
θ_{n−1}^{α} = c_{β}^{αρ}θ_{ρ}^{β}

4.5.3 The filtration introduced above has

¯F̄^{2} = {φ_{α_{1}...α_{p}K}^{I_{1}...I_{p}}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ dy^{K} ∧ dt^{n−1} ∧ dt^{n}}
¯F̄^{1} = \{φ′{I_{1}...I_{p} \atop α_{1}...α_{p}K}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ dy^{K} ∧ dt^{n−1} + φ′′{I_{1}...I_{p} \atop α_{1}...α_{p}K}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ dy^{K} ∧ dt^{n}\}
¯F̄^{0} = {all forms φ}

Then we have

\{ \begin{matrix}H^{q}( ¯F̄^{2}) = 0, & q ≠ n \\ H^{q}( ¯F̄^{1}/ ¯F̄^{2}) = 0, & q ≠ n − 1 \\ H^{q}( ¯F̄^{0}/ ¯F̄^{1}) = 0, & q ≠ n − 2 \end{matrix}

and the ¯ Ē spectral sequence reduces to the commutative diagram

0 ↓

H^{n−2}( ¯F̄^{0})

↓

H^{n−2}( ¯F̄^{0}/ ¯F̄^{1})  0
↓^{δ_{0}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} δ_{1}  ↓
0 → H^{n−1}( ¯F̄^{1})  ⟶  H^{n−1}( ¯F̄^{1}/ ¯F̄^{2}) ⟶^{δ_{2}} H^{n}( ¯F̄^{2}) → H^{n}( ¯F̄^{1}) → 0
↓  ↓
H^{n−1}( ¯F̄^{0})  H^{n}( ¯F̄^{0}) → 0
↓  ↓
0  0

In the split case δ_{0} = δ_{1} = δ_{2} = 0. This has the interpretation that a resolution of M lifts to one of Λ_{Δ}^{p}M. Thus

M has \{ \begin{matrix}generators w^{α} \\ generating relations r_{n}: e_{n}w^{α} = 0, r_{n−1}: e_{n−1}w^{α} = 0 \\ generating relation among the relations: e_{n−1}r_{n} − e_{n}r_{n−1} = 0 \end{matrix}

4.5.4 and when p = 2

Λ_{Δ}^{2}M has \{ \begin{matrix}generators φ = φ_{αβ}^{I}w^{α} ∧ w^{β}, φ + (−1)^{|I|}^{t}φ = 0 \\ generating relations r_{n} given by e_{n}φ = 0, r_{n−1} given by e_{n−1}φ = 0 \\ generating relations among the relations given by e_{n−1}r_{n} − e_{n}r_{n−1} = 0 \end{matrix}

corresponding to H^{n}( ¯F̄^{2}), H^{n−1}( ¯F̄^{1}/ ¯F̄^{2}), H^{n−2}( ¯F̄^{0}/ ¯F̄^{1}) respectively in the above diagram.

In the non-split case the groups H^{n}( ¯F̄^{2}),H^{n−1}( ¯F̄^{1}/ ¯F̄^{2}), H^{n−2}( ¯F̄^{0}/ ¯F̄^{1}) are the same and represent what we may call “postulated” generators, relations, and relations among the relations. The actual generators are obtained by reducing the postulated generators modulo image δ_{2}. The postulated relations are affected in two ways. First, they may not lift to actual relations – only those in kernel δ_{2} will do so. Secondly, those that do must be reduced by image δ_{1}. Finally, the postulated relations among the relations H^{n−2}( ¯F̄^{0}/ ¯F̄^{1}) may not lift – only ker δ_{1} = ker δ_{0} will do so. In all of this, however, we see that the minimal presentation of Λ_{Δ}^{2}M, and indeed of a general Λ_{Δ}^{p}M, has mappings of the same degree as in the split case. Thus Λ_{Δ}^{p}M is involutive.

4.5.5 4.6 The mixed case

We shall now illustrate how the general case goes by considering a tableau of the form

n_{1} + n_{2} = u r_{1} + r_{2} = s _{n_{2}} _{n_{1}} r_{2}{ r_{1}{

It is to be proved that

H^{q}(Λ_{Δ}^{p}M) = 0, 0 ≦ q < n_{1}.

In the preceding section we separated the independent variables (horizontal axis) by setting

(1)  \begin{aligned}y^{ρ} & = x^{ρ},  1 ≦ ρ, σ ≦ n_{1} \\ t^{λ} & = x^{λ},  n_{1} + 1 ≦ λ, μ ≦ n \end{aligned}

and filtering Ω^{∗} by the images ¯ F̄^{a} of

Ω^{∗} ⊗ Λ^{a}{dt^{n_{1}+1}, …, dt^{n}} ⟶ Ω^{∗+a}.

In this case we shall also separate the dependent variables (vertical columns) as indicated by the blocks in the tableau and shall separate the effect of differentiation with respect to the two sets of dependent variables. This method will work in general, and, as in the previous sections, the vanishing result does not require involutivity of the tableau, this being needed only to insure the involutivity of the Λ_{Δ}^{p}M. The main difficulty lies in organizing the notation.

Proceeding to the proof, we shall use the above ranges of indices and in addition shall let

\{ \begin{matrix}I, J, K & ⊂ [1, …, n_{1}] \\ R, S & ⊂ [n_{1} + 1, …, n] \end{matrix}

4.6.1 The symbol relations are

θ_{IR}^{α} ≡ 0 mod {θ_{J}^{α}, θ_{KS}^{ϵ}} for |R| > 0, α > r_{1},

corresponding to the lower right hand block α > r_{1}, i > n_{1} in the tableau. The forms θ_{I}^{α}, θ_{JR}^{ϵ} are independent and we look at a typical spanning exterior monomial

(2)  θ = θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{k}}^{α_{k}} ∧ θ_{J_{1}R_{1}}^{ϵ_{1}} ∧ … ∧ θ_{J_{ℓ}R_{ℓ}}^{ϵ_{ℓ}} ∧ dy^{K} ∧ dt^{S}

where all |R_{ν}| > 0. We set

¯w̄(θ) = |S|
¯F̄^{a} = {forms φ with ¯ w̄(φ) ≧ φ},

where this means that each monomial in the expression of φ has at least a dt^{λ′}s. Then we have a d-stable filtration

¯F̄^{n_{2}} ⊂ ¯ F̄^{n_{2}−1} ⊂ … ⊂ ¯ F̄^{1} ⊂ ¯ F̄^{0} = Ω^{∗}

and from

dθ_{I}^{α} ≡ −θ_{Iρ}^{α} ∧ dy^{ρ} mod {dt^{n_{1}+1}, …, dt^{n}}
dθ_{JR}^{ϵ} = −θ_{JρR}^{ϵ} ∧ dy^{ρ} − θ_{JRλ}^{ϵ} ∧ dt^{λ}

we see that on the successive quotients ¯ F̄^{a}/ ¯F̄^{a+1} the exterior derivative only “sees” dy^{1}, …, dy^{n_{1}}. More precisely, we let

¯Ω̄{k,ℓ \atop \underline{R},S} =\{span of the forms (2) where S with |S| = a and
\underline{R} = (R_{1}, …, R_{ℓ}) is fixed\}.

Then

¯F̄^{a}/ ¯F̄^{a+1} ≅ ⨁_{\begin{matrix}k+ℓ=p \\ \underline{R},S \end{matrix}} ¯ Ω̄{k,ℓ \atop \underline{R},S}.

4.6.2 If on ¯ Ω̄{k,ℓ \atop \underline{R},S} we set

d′θ = d_{y}(θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{k}}^{α_{k}}) ∧ θ_{J_{1}R_{1}}^{ϵ_{1}} ∧ … ∧ θ_{J_{ℓ}R_{ℓ}}^{ϵ_{ℓ}} ∧ dy^{K} ∧ dt^{S}
d′′θ = (−1)^{k}θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{k}}^{α_{k}} ∧ d_{y}(θ_{J_{1}R_{1}}^{ϵ_{1}} ∧ … ∧ θ_{J_{ℓ}R_{ℓ}}^{ϵ_{ℓ}}) ∧ dy^{K} ∧ dt^{S}

where

d_{y}θ_{I}^{α} = −θ_{Iρ}^{α} ∧ dy^{ρ}
d_{y}θ_{JR}^{ϵ} = −θ_{JρR}^{ϵ} ∧ dy^{ρ}

then the induced differential ¯ d̄ on ¯ F̄^{a}/ ¯F̄^{a+1} is

¯d̄= d′ + d′′.

Moreover,

d′: ¯ Ω̄{k,ℓ \atop \underline{R},S} → ¯ Ω̄{k+1,ℓ \atop \underline{R},S}
d′′: ¯ Ω̄{k,ℓ \atop \underline{R},S} → ¯ Ω̄{k,ℓ+1 \atop \underline{R},S}
d′^{2} = d′′^{2} = d′d′′ + d′′d′ = 0

The point is that both the d′-cohomology and the d′′-cohomology look like the unconstrained case, and consequently

H^{q}( ¯F̄^{a}/ ¯F̄^{a+1}) = 0, q < n_{1} + a.

This implies that

H^{q}(Λ_{Δ}^{p}M) = 0, q < n_{1}

as desired.

The general case will be clear from a three-step tableau

r_{2}{ r_{1}{  _{n_{3}} _{n_{2}} _{n_{1}}

4.6.3 We keep the same splitting of independent variables (1) and filtration ¯ F̄^{a} on Ω^{∗} by the number of dt^{λ}s. We shall use the ranges of indices

\{ \begin{matrix}1 ≦ ϵ ≦ r_{1}, 1 ≦ η ≦ r_{2} \\ 1 ≦ i, j ≦ n_{1}, n_{1} + 1 ≦ ρ, σ ≦ n_{1} + n_{2}, n_{1} + n_{2} + 1 ≦ λ, μ ≦ n \end{matrix}

and corresponding blocks

I = (i_{1}, …, i_{k}), R, S = (ρ_{1}, …, ρ_{ℓ}), M = (λ_{1}, …, λ_{m}).

The following are a basis for the 1-forms in the ideal

θ_{I}^{α}, θ_{IR}^{ϵ} where |R| > 0, θ_{IRM}^{η} where |M| > 0

and forms in ¯ F̄^{a}(F^{p}/F^{p+1}) are sums of monomials

(3) \underbrace{θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{k}}^{α_{k}}} ∧ \underbrace{θ_{I_{1}R_{1}}^{ϵ_{1}} ∧ … ∧ θ_{I_{ℓ}R_{ℓ}}^{ϵ_{ℓ}}} ∧ \underbrace{θ_{I_{1}S_{1}M_{1}}^{η_{1}} ∧ … ∧ θ_{I_{m}S_{m}M_{m}}^{η_{m}}} ∧ dy^{J} ∧ dt^{S} ∧

where k+ℓ+m = p, |S|+|N| ≧ a. As above we may split the induced exterior differentiation n ¯ F̄^{R}/ ¯F̄^{R+1} into

¯d̄= d′ + d′′ + d′′′

where d′, d′′, d′′′ are mutually graded-commuting and correspond to these vertical blocks in the above tableau. Each of d′, d′′, d′′′ does not effect the index blocks R_{1}, …, R_{ℓ}, S_{1}M_{1}, …, S_{m}M_{m}, and the individual cohomologies are all isomorphic to cohomology in the unconstrained case. Unwinding what this means yields the desired vanishing result.

4.6.4 5. Structure of the characteristic cohomology

We have proved that H̄^{q} = 0 for 0 < q < n − ℓ where the characteristic number ℓ measures degree of overdeterminedness of the exterior differential system. In this section we shall examine the structure of the remaining groups, focussing on the space 𝒞 = H̄^{n−ℓ} of conservation laws for the system. This discussion will require that the exterior differential system be involutive, and it will show that 𝒞 may be canonically identified with the kernel of a first order linear differential operator between naturally defined vector bundles.

One may ask: Is this rather abstract theorem of practical value for computing conservation laws in specific examples? A fair question, one which we shall begin to address when we turn to the computation of examples later in this paper and, especially, in Part II. 5.1 A variant of the Spencer complex

Let (X, ℐ) be the infinite prolongation of an involutive exterior differential system ℐ_{0} on a manifold X_{0}. The exterior powers of ℐ define a filtration F^{p}Ω^{∗}(X) whose associated graded we have denoted by Ω̄^{p,∗} with induced differential d̄. For p ≧ 1 there is then defined the (non-trivial) weight filtration W_{k}(Ω̄^{p,∗}) and we have denoted its associated graded by Ω̄{p,∗ \atop k} with induced differential δ. The basic property is that δ is linear over the functions, and making suitable constant rank assumptions we denote by ℋ_{k}^{p,q} the vector bundle over X defined by the cohomology of the complex \{Ω̄{p,∗ \atop k}, δ\}

ℋ_{k}^{p,q} = H^{q}(Ω̄{p,∗ \atop k}, δ).

We also set

ℋ^{p,q} = ⊕_{k} ℋ_{k}^{p,q}.

Now there is a spectral sequence E_{r}^{p,q} abutting to H^{∗}(X) and with

E_{1}^{p,q} = H^{q}(Ω̄^{p,∗}, d̄)
E_{1}^{0,q} = H̄^{q}  (our primary object of study).

The weight filtration then gives a second spectral sequence {Ē_{r}^{k,q}} abutting to E_{1}^{p,∗} for each fixed p > 0 and with

Ē{k,q \atop 1} = ℋ_{k}^{p,q}.

5.1.1 Our main result thus far has been to infer that H̄^{q} = 0 for 0 < q < n − ℓ by showing that (i) ℋ^{p,q} = 0 for p > 0, q < n − ℓ, which implies (ii) E_{1}^{p,q} = 0 for p > 0, q < n − ℓ, and then this implies the above stated vanishing result for the H̄^{q}. We will now study some aspects of the structure of the non-zero part of the characteristic cohomology. To state the result, we denote by

(1)  ∇_{k}: ℋ_{k}^{p,q} → ℋ_{k−1}^{p,q+1},  p > 0

the differential in the Ē_{r}^{k, q} spectral sequence, and we denote by

∇: ℋ^{p,q} → ℋ^{p,q+1}

the direct sum over k of the mappings (1). Then ∇^{2} = 0 and the cohomology of the complex {ℋ^{p,∗}, ∇} computes the E_{2}-term of the weight spectral sequence for each p > 0. Our result is given by the

Theorem: (i) In the involutive case

E_{1}^{1,q} = H^{q}(ℋ^{1,∗}, ∇).

(ii) For all p ≧ 1,

E_{1}^{p,n−ℓ} ⊂ ℋ^{p,n−ℓ}

is the subspace given by the kernel of a canonical linear differential operator.

Corollary: 𝒞 ≅ ker ∇ ∩ ker d_{1} where ∇: ℋ^{1,n−ℓ} → ℋ^{1,n−ℓ+1} and d_{1}: E_{1}^{1,n−ℓ} → E_{1}^{2,n−ℓ} and canonically defined 1^{st} order linear differential operators.

Proof: We have seen in Section 4.1 above (see equation (1) there) that ℋ_{k}^{p,q} is the vector bundle whose typical fiber at a point ξ is

H_{k+q}^{q}(Λ_{Δ}^{p}M_{ξ})

where M_{ξ} is the graded module associated to the symbol of the exterior differential system at the point ξ, and where H_{m}^{q}( ·) refers to (the dual of) Koszul homology in module degree

5.1.2 m. The index k + q appears above and the weight filtration encodes both the module degree and the exterior degree.

When p = 1, since M is involutive and is generated in degree zero, we have that

H_{+}^{q}(M) = 0.

Thus the weight spectral sequence degenerates at the E_{2}-term, and

Ē_{2} = H^{∗}(Ē_{1}, ∇)

Unwinding the notation gives the first result.

The remaining part of the theorem follows from general properties of spectral sequences \{Ē{k,q \atop r}\} with Ē{k,q \atop 1} = 0 for 0 ≦ q < n − ℓ. Q.E.D.

Discussion: The complex {ℋ^{1,∗}, ∇} will be called the universal Spencer complex. We shall explain this point.

Referring to the notations introduced in Section 2.1, in any local coordinate system (x^{i}, u^{α}, p_{i}^{α}, p_{ij}^{α}, …) on G_{n,∞}(X_{0}) there is a splitting

d = d_{V} + d_{H}

of the exterior derivative into “vertical” and “horizontal” components. Explicitly,

d_{V} = ∑_{|I|=0}^{∞} ∂/∂p_{K}^{α}θ_{K}^{α}

where we have set p_{0}^{α} = u^{α}

\{ \begin{matrix}d_{H} & = ∑ D_{i}dx^{i} \\ D_{i} & = ∂/∂x^{i} + ∑_{|I|=0}^{∞} ∂/∂p_{I}^{α}p_{Ii}^{α}. \end{matrix}

For any integral manifold

f: N → G_{n,∞}(X_{0})

5.1.3 of the universal Pfaffian system θ_{I}^{α} = 0, |I| ≧ 0, on X, we have

f^{∗}d_{V} θ_{I}^{α} = 0
f^{∗}d_{H}θ_{I}^{α} = 0.

In fact, if the integral manifold is given by

(x^{i}) → (x^{i}, u^{α}(x), p_{i}^{α}(x), p_{ij}^{α}(x), …, ),

then for any function φ on X

f^{∗}(D_{i}φ) = ∂/∂x^{i}(f^{∗}φ),

so that D_{i} is a “universal form” of ∂/∂x^{i}.

Let now X ⊂ G_{n,∞}(X_{0}) be the underlying manifold of infinite prolongation ℐ of an involutive exterior differential system ℐ_{0} on X_{0}. We have seen that the equations defining X are

D_{I}F = 0

where F (x^{i}, u^{α}, p_{i}^{α}) varies over all the functions on G_{∗}(T X_{0}) defined by θ ∈ ℐ_{0}^{n}, i.e.

F (E) = θ|_{E}

for E ∈ G_{n}(T X_{0}). It follows that the vector fields D_{i}|_{X} are tangent to X and induce vector fields on X that we still denote by D_{i}. In these local coordinates the operators

∇: ℋ^{1,q} → ℋ^{1,q+1}

are then easily seen to be horizontal, in the sense that

∇ = ∑ A^{i}D_{i} + B

where A^{i}, B ∈ Hom(ℋ^{1,q}, ℋ^{1,q+1}) are bundle maps that are linear over the functions. It follows that {ℋ^{1,∗}, ∇} induces on any integral manifold

f: N → X_{0}

5.1.4 of ℐ_{0} a complex of vector bundles and 1^{st} order linear operators, which we shall denote by {ℋ_{f}^{∗}, ∇_{f}}. In fact, it may be seen that {ℋ_{f}^{∗}, ∇_{f}} is just the dual of (a suitable relabelling of) the Spencer complex associated to the PDE system which is defined by the exterior differential system ℐ_{0} in our particular local coordinate system (cf. Chapters IX, X of [2] for a discussion of the Spencer complex and further references).

One may invert this discussion. That is, not knowing what Spencer cohomology and the Spencer complex are, one is led naturally and inevitably to them once one agrees to study the characteristic cohomology of an exterior differential system.

The first term in the Spencer complex is

(2)  ∇_{f}: ℋ_{f}^{n−ℓ} → ℋ_{f}^{n−ℓ+1},

and it is an interesting point that the universal conservation laws appear naturally as a subspace of ker ∇_{f} in (2).

What the theorem and its corollary give is that the space of conservation laws is given by the kernel of a canonical linear differential operator, modulo nothing. We have seen a special case of this in the contact and symplectic manifold examples in Section 1. In the case of determined PDE systems, this phenomena has been formulated in coordinates and used to compute examples by Vinogradov [15] and others. As we shall see in Part II of this paper, the result itself is of some use, but most importantly it serves to guide the generally complicated computations that arise in non-trivial applications of E. Cartan’s equivalence method.

5.1.5 5.2 Quadratic conservation laws for constant coefficient linear equations

As a first application of the general theory in the preceding section, we shall investigate the space of conservation laws associated to a constant coefficient linear PDE system. We first consider the determined case which in suitable coordinates may be written as

(1)  \frac{∂u^{α}(x)}{∂x^{n}} = b_{β}^{αρ} \frac{∂u^{β}(x)}{∂x^{ρ}}, 1 ≤ ρ, σ ≤ n − 1

where the b^{ρ} = ∥b_{β}^{αρ}∥ are constant matrices. Associated to this system is its symbol and we denote by M the graded module associated to the symbol. Denoting as usual by (x^{i}, u^{α}, p_{i}^{α}, p_{ij}^{α}, …) the natural coordinates on the space J^{∞}(ℝ^{n}, ℝ^{s}) of infinite jets of mappings from ℝ^{n} to ℝ^{s}, we observe that the group of gauge transformations that leave invariant the property that (1) be a linear, homogeneous constant coefficient equation will in general be equal to GL_{n} × GL_{s}, and it is this symmetry group that we shall take. We are interested in conservation laws that are invariant under translation in ℝ^{n}, and for reasons to be explained below we shall focus on those represented in E_{1}^{1,n−1} by translation invariant forms whose coefficients are linear in the jet coordinates u^{α}, p_{i}^{α}, p_{ij}^{α}, …. The following will also be explained below:

Definition: Conservation laws represented in E_{1}^{1,n−1} by translation invariant forms whose coefficients are linear in the jet coordinates will be called quadratic conservation laws for the PDE system (1).

Theorem: The space of quadratic conservation laws for (1) is naturally isomorphic to

H^{n−1}(Sym_{Δ}^{2}M)

Thus, the Δ-multilinear algebra appears once again, in a different form, in a natural way associated to a geometric problem. Proof: We shall use the notations in the preceding section. For a point ξ ∈ X we shall successively compute that part of

(i) ℋ_{ξ}^{1,n−1} (ii) ∇: ℋ^{1,n−1} → ℋ^{1,n}

5.2.1 (iii) d_{1}: ker ∇ → ℋ^{2,n−1} corresponding to forms that are translation invariant and linear in the jet coordinates. By the corollary in the preceding section, the kernel of (iii) is identified with the space of conservation laws for (1).

Step One:

We first recall notations.

• dx^{i} and θ_{I}^{i} are linear functions on two vector spaces, the dx^{i} being linearly independent and the θ_{I}^{α} subject to the relations

b_{α}^{λi}θ_{Ii}^{α} = 0

• Ω^{p,q} is spanned by the forms

θ_{I_{1}}^{α_{1}} ∧ … ∧ θ_{I_{p}}^{α_{p}} ∧ dx^{j_{1}} ∧ … ∧ dx^{j_{q}}

and Ω^{∗} = ⊕Ω^{p,q}

• The differential

d: Ω^{p,q} → Ω^{p,q+1}

is defined by d(dx^{i}) = 0 and

dθ_{I}^{α} = −θ_{Ii}^{α} ∧ dx^{i},

together with the usual properties of being a derivation of the exterior algebra

• We will denote by Λ_{m}^{∗} = ⊕Λ_{m}^{p,q} a complex analogous to Ω^{p,q} but where the number of dx^{i}’s is m and where there are no relations on the θ_{I}^{α} – this “free” case models the unconstrained geometric situation in which the number of independent variables is m, and we have seen that

H^{k}(Λ_{m}^{∗}) = 0, k < m.

In order to compute the fiber

ℋ_{ξ}^{1,n−1} ≅ H^{1,n−1}(Ω^{∗}, d)

5.2.2 we proceed as in Section 4.4 and set

Ω′^{p,q} = dx^{n} ∧ Ω′^{p,q−1} ≅ Λ_{n−1}^{p,q−1}[1]
Ω′′^{p,q} = Ω^{p,q}/Ω′^{p,q} ≅ Λ_{n−1}^{p,q}.

In the exact cohomology sequence

0 → H^{1,n−1}(Ω^{∗}) → H^{1,n−1}(Ω′′^{∗}) ⟶^{δδδ} H^{1,n}(Ω′^{∗})

we have δδδ = 0 as noted in Section 4.4. In fact, given

φ′′ = f_{α}θ^{α} ∧ dx_{n} ∈ H^{1,n−1}(Ω′′^{∗})

we see that

φ = f_{α}θ^{α} ∧ dx_{n} − f_{β}b_{α}^{βρ}θ^{α} ∧ dx_{ρ}

satisfies

δφ = 0

and thus gives a class φ ∈ H^{1,n−1}(Ω^{∗}) which serves to provide a lifting of the class defined by φ′′ ∈ H^{1,n−1}(Ω′′^{∗}).

Step two: We will consider sections

φ = f_{α}θ^{α} ∧ dx_{n} − f_{β}b_{α}^{βρ}θ^{α} ∧ dx_{ρ}

of ℋ^{1,n−1} where f_{α} is a function on X. In fact, we shall consider functions of the form

f_{α} = ∑_{|I|≥0, β} f_{αβ}^{I}p_{I}^{β}

that are linear in the fiber coordinates, and where we set u^{α} = p_{0}^{α}. As noted above, this has intrinsic meaning, and by a scaling argument in the independent variables x^{i} we see that it will suffice to consider the case where

f_{α} = ∑_{|I|=k, β} f_{αβ}^{I}p_{I}^{β}

5.2.3 is homogeneous of degree −k in the dependent variables. We want to compute

∇: ℋ^{1,n−1} → ℋ^{1,n}

for φ ∈ ℋ^{1,n−1} of this form. By definition

∇φ = dφ mod ℐ ∧ ℐ
= (D_{n}f_{α} − D_{ρ}f_{β}b_{α}^{βρ}) θ^{α} ∧ dx
= (f_{αβ}^{I}p_{In}^{β} − f_{βγ}^{I}p_{Iρ}^{γ}b_{α}^{βρ}) θ^{α} ∧ dx
= (f_{αβ}^{I}b_{γ}^{βρ} − b_{α}^{βρ}f_{βγ}^{I}) p_{Iρ}^{γ}θ^{α} ∧ dx.

If we set

φ(ξ) = ∥f_{αβ}(ξ)∥ = ∥f_{αβ}^{I}ξ_{I} ∥
b(ξ) = ∥b_{α}^{βρ}ξ_{ρ}∥

then we have

(2)  ∇φ = 0 ⟺ [φ(ξ), b(ξ)] = 0.

We note the analogy with the proposition in Section 4.4, where the condition [φ(ξ), b(ξ)] = 0 arose in a different context. We note there the symmetry condition

φ + (−1)^{|I|}^{t}φ = 0

that was imposed on φ.

Step three: We assume that

φ = f_{αβ}^{I}p_{I}^{β}θ^{α} ∧ dx_{n} − f_{βγ}^{I}p_{I}^{γ}b_{α}^{βρ}θ^{α} ∧ dx_{ρ} ∈ E_{1}^{1,n−1}

satisfies ∇φ = 0, and we want to compute the condition

d_{1}φ = 0 in E_{1}^{2,n−1}.

5.2.4 For this we denote by E′_{1}, E′′_{1} the spectral sequences associated to the bigraded complexes Ω′^{∗}, Ω′′^{∗} respectively. Then we have a diagram

0 ⟶ E_{1}^{2,n−1} ⟶ E′′{2,n−1 \atop 1}
↑^{d_{1}} ↑^{d′′_{1}}
0 ⟶ E_{1}^{1,n−1} ⟶ E′′{1,n−1 \atop 1},

from which we see that

d_{1}φ = 0 ⟺ d′′_{1}φ′′ = 0

where

φ′′ = f_{αβ}^{I}p_{I}^{β}θ^{α} ∧ dx_{n}

is the image in E′′{1,n−1 \atop 1} of φ ∈ E_{1}^{1,n−1}. Now

d′′_{1}φ′′ = f_{αβ}^{I}θ_{I}^{β} ∧ θ^{α} ∧ dx_{n}.

Setting f_{αβ}^{I} = f̂{I \atop αβ} + ˆ f̂{I \atop αβ} where

f̂{I \atop αβ} = \frac{1}{2} (f_{αβ}^{I} + (−1)^{|I|}f_{βα}^{I})
ˆf̂_{{I \atop αβ}} = \frac{1}{2} (f_{αβ}^{I} − (−1)^{|I|}f_{βα}^{I})

we see from equation (3) in Section 3.2 that

d′′_{1}φ′′ = 0 ⟺ f̂{I \atop αβ} = 0.

Thus

(3)  \{∇φ = d_{1}φ = 0\} ⇔ \{ \begin{matrix}φ = f_{αβ}^{I}p_{I}^{β}θ^{α} ∧ dx_{n} − f_{βγ}^{I}p_{I}^{α}b_{α}^{βρ}θ^{α} ∧ dx_{ρ} \\ where [φ(ξ), bξ)] = 0 and \\ φ(ξ) − (−1)^{|I|}^{t}φ(ξ) = 0 \end{matrix}\}

By (3) in Section 3.2 the space of φ satisfying the condition

φ(ξ) − (−1)^{|I|} ^{t}φ(ξ) = 0

5.2.5 is isomorphic to H^{n−1}(Sym_{Δ}^{2}M′) where M′ is the free module over S′ = 𝔽[e_{1}, …, e_{n−1}] with generators w^{1}, …, w^{s}. It remains to identify those φ satisfying both conditions on the right hand side of (3) with H^{n−1}(Sym_{Δ}^{2}M). This is done in essentially the same way as for Λ_{Δ}^{2}M′, and we will only sketch the argument.

In general, to compute the H^{q}(Sym_{Δ}^{p}M) we consider the complex Ω̃^{∗} = ⊕Ω̃^{p,q} of forms

ψ = ψ_{α_{1}...α_{ρ}J}^{I_{1}...I_{p}}θ_{I_{1}}^{α_{1}} ∘ ⋯ ∘ θ_{I_{p}}^{α_{p}} ⊗ dx^{J}, |J| = q,

where the θ_{I_{ν}}^{α_{ν}} are independent forms multiplied symmetrically. We define a differential δ by linearity and

δ (θ_{I_{1}}^{α_{1}} ∘ ⋯ ∘ θ_{I_{p}}^{α_{p}}dx^{J}) = ∑_{ν} θ_{I_{1}}^{α_{1}} ∘ ⋯ ∘ (δθ_{I_{ν}}^{α_{ν}} ) ∘ ⋯ ∘ θ_{I_{p}}^{α_{p}} ⊗ dx^{J}

δ (θ_{I}^{α}) = θ_{Iρ}^{α} ⊗ dx^{ρ} + b_{β}^{αρ}θ_{Iρ}^{β} ⊗ dx^{n}, 1 ≤ ρ ≤ n − 1,

where we move multiplication by dx^{j} across the θ_{I_{μ}}^{α_{μ}} and exteriorly multiply it with dx^{J}. Thus, more or less by definition, the cohomology of this complex at the place with p θ’s and q dx^{j}’s computes H^{q}(Sym_{Δ}^{p}M).

We next set Ω̃′^{∗} = dx^{n}∧Ω̃^{∗} and Ω̃′′^{∗} = Ω̃^{∗}/Ω̃′^{∗}, and note as in the case of Λ_{Δ}^{p}M we have

\{ \begin{matrix}H^{q}(Ω̃′^{∗}) = 0,  q < n \\ H^{q}(Ω̃′′^{∗}) = 0,  q < n − 1. \end{matrix}

From the exact cohomology sequence

0 → H^{n−1}(Ω̃^{2,∗}) → H^{n−1}(Ω̃′′^{2,∗}) ⟶^{δ̃} H^{n}(Ω̃′^{2,∗})

for a class in H^{n−1}(Ω̃′′^{2,∗}) = H^{n−1}(Sym_{Δ}^{2}M′) represented by

φ̃′′ = f_{αβ}^{I}θ_{I}^{α} ∘ θ^{β} ⊗ dx_{n}, f_{αβ}^{I} − (−1)^{|I|}f_{βα}^{I} = 0

we have as in the proposition in Section 4.4 that

δφ̃′′ = 0 ⟺ [φ(ξ), b(ξ)] = 0.

Since ker δ ≅ H^{n−1}(Sym_{Δ}^{2}M) we have completed the proof of the theorem.

5.2.6 Discussion: the conservation laws 𝒞 = E_{1}^{0,n−1} map injectively to E_{1}^{1,n−1} by

d_{1}: E_{1}^{0,n−1} → E_{1}^{1,n−1}.

The above theorem identifies the classes in E_{1}^{1,n−1} that are in the image of d_{1} and that are linear in the jet coordinates. But since d_{1} is determined by the rules

d_{1}(θ_{I}^{α}) = d_{1}(dx^{j}) = 0
d_{1}(f) = ∑ ∂f/∂p_{I}^{α}θ_{I}^{α}

we see that d_{1} maps exactly those classes in E_{1}^{0,n−1} that are quadratic in the p_{I}^{α}’s to classes in E_{1}^{1,n−1} that are linear in these coordinates. This explains the above definition.

Now, in the classical calculus of variations perhaps the most interesting Lagrangians are those of the form

L(x^{i}, u^{α}, p_{i}^{α}, …, p_{I}^{α})dx, |I| ≦ k,

which are quadratic in the highest order terms. Similarly, as illustrated by the various classical conservation laws and the KDV, we may expect conservation laws that are quadratic in their highest order terms to be of particular interest. It is for this reason that we have focussed on the quadratic conservation laws in the above theorem. We note that finding these indirectly by studying classes in E_{1}^{1,n−1} that are linear in the p_{I}^{α} and satisfy certain conditions appears to be considerably more efficient than directly studying the quadratic conservation laws.

5.2.7 6. Global Results

The results obtained thus far are microlocal, in a precise sense to be explained below. In this final section we shall give a global theorem in cohomology, one which follows in a more or less standard way from our local result, and then we shall discuss a more geometric dual set of questions which pertain to a very prescient theorem of Thom [10]. 6.1 A relation between ordinary and characteristic cohomology

Let ℐ_{0} be an involutive exterior differential system on a connected manifold X_{0}, and as above denote by (X_{k}, ℐ_{k}) the k-th prolongation with X = \underleftarrow{lim} X_{k},and ℐ = ℐ_{0} ⋃_{k>0} ℐ_{k} (cf. Section 1.3). Since ℐ_{0} may not be a quasi-linear Pfaffian differential system, we need to be a little careful about what is meant that ℐ_{0} be involutive. Denote by G_{n}(ℐ_{0}) ⊂ G_{n}(T X_{0}) the set of n-dimensional integral elements of ℐ_{0}. A priori, G_{n}(ℐ_{0}) is locally the locus of common zeros of a collection of smooth functions defined in an open set of G_{n}(T X_{0}) (these functions are of the special form given in Section 1.3, but that only serves to restrict somewhat the singularities of G_{n}(ℐ_{0}) in the fibers of G_{n}(T X_{0}) → X_{0}). We shall make the following regularity assumptions:

There exists a connected, locally closed submanifold X_{1} of G_{n}(ℐ_{0}) such that

(i) Each point E ∈ X_{1} has a neighborhood U in G_{n}(T X_{0}) such that U ∩X_{1} = U ∩G_{n}(ℐ_{0}) and such that the forms in ℐ_{0} give a regular set of defining equations for U ∩ X_{1},^{(1)}

(ii) X_{1} consists of ordinary integral elements, and (iii) the projection X_{1} → X_{0} is a submersion. The concept that E ∈ G_{n}(ℐ) be ordinary is also explained in Chapter III of [2]; in the real analytic case it implies that there is a local integral manifold of ℐ_{0} that is tangent to E. It is also shown there that with the assumptions (i)-(iii) the prolongation tower (X_{k}, ℐ_{k}) exists and satisfies the conditions of Section 1.3 above. In particular, for k ≧ 1 the submersions X_{k+1} → X_{k} are fibrations whose fibers are affine linear spaces, and thus the projections X_{k} → X_{1} are all homotopy equivalences. (For more detail, the reader may want to consult Chapter VI of [2] as well.)

6.1.1

(1)In Chapter III of [2] there is an explanation of how forms on X_{0} give functions on G_{n}(T X_{0}) and what it means for a set of such functions to give a regular set of defining equations.

By microlocal we shall mean in a sufficiently small neighborhood in X_{1} of an ordinary integral element E ∈ X_{1}. Our main vanishing result, given by Theorem 2 in Section 4.2, states that in the appropriate range the characteristic cohomology vanishes microlocally, i.e. it vanishes in the range 0 < q < n − ℓ in open sets π_{1}^{−1}(U) ⊂ X where π_{1}: X → X_{1} is the projection and U ⊂ X_{1} is a sufficiently small open set. We shall now prove Theorem 1: Let ℐ_{0} be an involutive exterior differential system of character ℓ and with trivial infinite derived system.^{(2)} Then we have

(i) H^{q}(X, ℝ) ≅ H̄^{q}(X), 0 ≦ q < n − ℓ (ii) 0 → H^{n−ℓ}(X, ℝ) → H̄^{n−ℓ}(X) → H^{0}(ℋ̄^{n−ℓ}) → H^{n−ℓ+1}(X, ℝ) → 0 where ℋ̄^{q} is the q-th characteristic cohomology sheaf on X. Proof: We recall that the topology on X is defined by taking the open sets to be π_{k}^{−1}(U) where U ⊂ X_{k} is an open set and π_{k}: X → X_{k} is the projection. Then we define the sheaves \underline{Ω}^{q}, \underline{ℐ}^{q}, \overline{\underline{Ω}}^{q} = \underline{Ω}^{q}/\underline{ℐ}^{q} by considering the appropriate smooth forms in the open sets π_{k}^{−1}(U). We then have a complex of sheaves over X

\overline{\underline{Ω}}· = \{\overline{\underline{Ω}}^{0} ⟶^{d̄} \overline{\underline{Ω}}^{1} ⟶^{d̄} ⋯ ⟶^{d̄} \underline{Ω}^{q} ⟶^{d̄} \underline{Ω}^{q+1} ⟶^{d̄} ⋯ \} ,

and we denote by ℋ̄^{q} the cohomology sheaves. By the assumption that ℐ_{0} has a trivial infinite derived system, it follows that

ℋ̄^{0} ≅ ℝ

is the constant sheaf. Our microlocal vanishing theorem gives

ℋ̄^{q} = 0  for  0 < q < n − ℓ.

The results (i), (ii) are then a standard consequence of these two facts.

In more detail, denote by ℍ(\overline{\underline{Ω}}·) the hypercohomology of the complex of sheaves \overline{\underline{Ω}}·. Then there are two spectral sequences ^{‵}E and ^{‵‵}E each abutting to ℍ(\overline{\underline{Ω}}·), one where

‵E^{p,q} = H_{δ}^{p}(ℋ̄^{q})

6.1.2 and the other where

(2)This means that there are locally no non-constant functions f with df ∈ ℐ_{0}—geometrically, it means that locally there are no “inessential variables” in the system.

‵‵E_{1}^{p,q} = Hq ; d̄(H_{δ}^{p}(\overline{\underline{Ω}}·)).

Here, δ is used to denote the Čech cohomology of the sheaf in question. But, by the usual partition of unity argument, H_{δ}^{p}(\overline{\underline{Ω}}·) = 0 for p > 0 and so

‵‵E_{1}^{p,q} = \{ \begin{matrix}0 & p > 0 \\ H̄^{q}(X) & p = 0 . \end{matrix}

On the other hand, ^{‵}E_{1}^{p,q} = 0 for 0 < q < n − ℓ, and the “picture” of the ^{‵}E_{1} is

.^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{•}  ^{•}  ^{•}  ^{•} H^{1}(X)  H^{n−ℓ}(X) H^{n−ℓ+1}(X) H^{0}(ℋ̄^{n−ℓ})  d_{n−ℓ}

The result follows from this. Q.E.D.

6.1.3 6.2 ℐ-homology

Let 𝒥 be an exterior differential system on a manifold Y. We let Δ^{q} denote a standard q-simplex and consider mappings

f: Δ^{q} ⟶ Y

that are of class C^{1} on Δ^{q}, including on the boundary, and that satisfy

f^{∗}θ = 0, θ ∈ 𝒥.

We will call such a simplex a 𝒥-simplex, and we may then define a 𝒥-chain to be a singular chain each of whose simplices is a 𝒥-simplex. The boundary of a 𝒥-chain is again an 𝒥-chain, and we denote by

H_{∗,𝒥}(Y)

the resulting homology. Definition: We shall call H_{∗,𝒥}(Y) the 𝒥-homology of the exterior differential system 𝒥.

This concept has been introduced by Thom [10] in the special case when Y is an open set of J^{k}(ℝ^{n}, ℝ^{s}) and 𝒥 is the restriction to Y of the contact system. He proved a result which is in a certain sense dual to the 1-line theorem of Vinogoadov (i.e., Vinogrodov’s result [13] in the unconstrained case), although it neither implies nor is implied by Vinogrodov’s theorem. We will formulate and discuss a possible extension of Thom’s result. It must be emphasized that the following discussion is highly speculative and is not intended to be anything more than suggestive. Apparently, there are great technical difficulties to be overcome before an actual proof of any of these results can be constructed. We have included this discussion here mainly because it indicates a geometric interpretation of the characteristic cohomology which we believe will have great importance for the theory.

First, remark that denoting now by

H_{𝒥}^{∗}(Y) = H^{∗}(Ω^{∗}(Y)/𝒥, d̄)

the characteristic cohomology of the complex (Ω^{∗}(Y)/𝒥, d̄), there is a natural pairing

(1)  H_{𝒥}^{q}(Y) ⊗ H_{q,𝒥}(Y) → ℝ.

6.2.1 A natural question then is whether the 𝒥-de-Rham Theorem holds. In other words, is the pairing (1) non-degenerate? We will see that, although in general the answer appears to be no, the question presents some very interesting and highly non-trivial issues.

There is always a natural mapping

H_{q,𝒥}(Y) → H_{q}(Y)

(homology with ℤ-coefficients), and by analogy with Theorem 1 in Section 6.1 above, we may pose the

Question: In case ℐ_{0} is an involutive exterior differential system of character ℓ with trivial infinite derived system, do we have

(2)  \{ \begin{matrix}(i) & H_{q,𝒥}(X) →^{∼} H_{q}(X) & 0 ≦ q < n − ℓ \\ (ii) & H_{n−ℓ,𝒥}(X) → H_{n−ℓ}(X) → 0? \end{matrix}

Thom’s result is that the answer to the same question for Y an open subset of J^{k}(ℝ^{n}, ℝ^{s}) is affirmative, so that we have

H_{q,𝒥}(Y) →^{∼} H_{q}(Y),  0 ≦ q < n
H_{n,𝒥}(Y) → H_{n}(Y) → 0.

We will indicate how the type of arguments Thom used might be used to establish (i) and (ii) above in the case where ℐ_{0} satisfies the condition that it “has no real Cartan characteristics in the range 0 ≦ q < n − ℓ”, a condition that we shall explain below and which is satisfied for the contact systems.

A technical issue that arises concerns the level of smoothness that we require of 𝒥-chains. Thus consider the induced mappings

(3)  H_{q,ℐ_{k+1}}(X_{k+1}) → H_{q,ℐ_{k}}(X_{k}), k ≧ 1.

These mappings are all isomorphisms in ordinary homology and H_{∗}(X) ≅ H_{∗}(X_{k}) for all k ≧ 1. If we have a piecewise-C^{j} ℐ_{k}-chain Γ_{k} on X_{k}, then its prolongation Γ_{k}^{∗} is a piecewise C^{j−1} chain on X_{k+1}. But unless j ≧ 2 we cannot say that Γ_{k}^{∗} will be a ℐ_{k+1}-chain. Moreover,

6.2.2 even if j ≧ 2 and Γ_{k} is an ℐ_{k}-cycle on X_{k}, it may not be the case that Γ_{k}^{∗} is an ℐ_{k+1}-cycle since the edges of Γ_{k} may separate under prolongation. Although we see no real reason why (3) should fail to be isomorphisms for all q, we shall not pursue this matter here and shall restrict our attention to X_{1}. Similar arguments to those below will apply to the other X_{k}.

Next, we point out that the question (2) is essentially local. That is, suppose that for sufficiently small neighborhoods U of each point in X_{1} we have for the reduced ℐ_{1}-homology groups

(4)  H̃_{q,ℐ_{1}}(U) = 0,  0 ≦ q < n − ℓ.

Then by standard local-to-global arguments such as those used in the preceeding section, we may infer (2) with X_{1} replacing X.

To discuss (4) it will be notationally more convenient to use cubes, denoted by □^{q} or simply □ if the dimension is understood, rather than simplices. To establish (4) the following sort of result must be proved:

(5) Let 0 ≦ q < n − ℓ and Σ ⊂ ∂□^{q+1} a set of q-dimensional faces of the boundary of a

(q+1)-cube. Suppose that

f: Σ → U

is an ℐ_{1}-integral manifold that has an extension as a piecewise-C^{1} mapping to all of □^{q+1}. Then f may be extended to a piecewise-C^{1} ℐ_{1}-integral manifold.^{1}

By arguments of the type used by Thom, the crucial case is when Σ = ∂□^{q+1} and this is the situation that we shall discuss.

Thus, let 0 ≦ q < n − ℓ and

(6)  f: ∂□^{q+1} → U

be a piecewise-C^{1} ℐ_{1}-integral manifold where U is a sufficiently small neighborhood of a point in X_{1}. We shall sketch arguments to the effect that:

6.2.3 (i) In case (6) is in general position, f may be extended to a piecewise-C^{1} ℐ_{1}-integral

1This means that □ may be subdivided and a continuous mapping f exists which is a C^{1} integral manifold on each of the sub-cubes.

manifold.

(ii) In case there are no real Cartan characteristics, we may deform f to be in general position. The essential case to understand (i) is when s_{0} = s_{1} = s_{2} > s_{3} ≧ 0. We shall take

first the case s_{3} = 0 (and n = 3), and then indicate the modifications necessary in general.

Locally we may choose coordinates (x, y, z, u, p, q, r) in J^{1}(ℝ^{3}, ℝ^{s}) such that the constant system is, as usual,

θ = du − pdx − qdy − rdz = 0

where u, p, q, r are ℝ^{s}-valued, and such that X_{1} is locally given by an equation

(7)  r = F (x, y, z, u, p, q)

where F is defined in convex open set in ℝ^{3+3s}. The integers n and ℓ are 3 and 1, respectively, and thus the relevant dimension range is 0 ≦ q < 2.

Suppose first that q = 0 and ∂I^{1} is the two points Υ_{i} = (x_{i}, y_{i}, z_{i}, u_{i}, p_{i}, q_{i}, r_{i}), i = 0, 1, and where r_{i} = F (x_{i}, y_{i}, z_{i}, u_{i}, p_{i}, q_{i}). We want to find a piecewise-C^{1} arc γ(t) with γ(0) = γ_{0}, γ(1) = γ_{1} and satisfying

(8)  \{ \begin{matrix}r(t) & = F (x(t), y(t), z(t), u(t), p(t), q(t)) \\ u′(t) & = p(t)x′(t) + q(t)y′(t) + r(t)z′(t). \end{matrix}

Given x(t), y(t), z(t), u(t) these are a set of finite (i.e., not differential) equations for p(t), q(t), r(t). Moreover, these finite equations are underdetermined. Thus we may expect that in general – i.e., when suitable open general position conditions are satisfied – we may solve these equations.

In fact, in the case q = 0 we may always solve the equations (8). If (x_{0}, y_{0}, z_{0}) ≠ (x_{1}, y_{1}, z_{1}) then we may assume that x_{0} ≠ x_{1} and y_{0} = y_{1}, z_{0} = z_{1}. We may then take an arbitrary arc x(t) with x(0) = x_{0}, x(1) = x_{1}, x′(t) ≠ 0, and set y(t) = z(t) = 0. We then choose a function u(t) with u(0) = u_{0}, u(1) = u_{1}, u′(0) = p_{0}x′_{0}, u′(1) = p_{1}x′_{1}

6.2.4 and then set p(t) = u′(t)/x′(t). Finally we may choose q(t) to be any arc with q(0) = q_{0}, q(1) = q_{1} to have a solution to (8). Of course, we must take care to insure that (x(t), y(t), z(t), u(t), p(t), q(t)) remains in the domain of definition of F, and this may be done by subdividing the interval and using a piecewise-C^{1} arc.

If, on the other hand, (x_{0}, y_{0}, z_{0}) = (x_{1}, y_{1}, z_{1}) then a similar argument – in fact with the same notation – applies.

Remark that in general the case q = 0 of (4) follows from Chow’s theorem, and indeed one may view (4) as an extension of that result.

More interesting is the case q = 1. If □ is the unit square 0 ≦ s ≦ 1, 0 ≦ t ≦ 1 in the (s, t) plane, then we must find a mapping

(9)  (s, t) → (x(s, t), y(s, t), z(s, t), u(s, t), p(s, t), q(s, t), r(s, t))

satisfying

(10)  \{ \begin{matrix}(i) & r & = F (x, y, z, u, p, q) \\ (ii) & u_{s} & = px_{s} + qy_{s} + rz_{s} \\ (iii) & u_{t} & = px_{t} + qy_{t} + rz_{t} , \end{matrix}

and where (9) is given on the boundary of the square such that (10) is satisfied there. If x(s, t), y(s, t), z(s, t), u(s, t) are given arbitrarily, then (10) is a determined set of finite equations. Equations (10)-(ii) and (10)-(iii) are analogous to the equations encountered in Thom’s result, and we may seek to adapt his arguments to our situation. In fact, this may be done provided that the boundary conditions are in general position. Thus, assume that the image of ∂□ is in general position in the sense that we may choose coordinates so that the projection of ∂□ to (x, y, z) space is the unit square in the x, y plane. Then we may replace s, t by x, y and seek to determine a mapping of the special form

(x, y) → (x, y, 0, u(x, y), p(x, y), q(x, y), r(x, y))

such that (10) is satisfied. Equations (10)-(ii) and (10)-(iii) reduce to those in Thom’s theorem, with no equations involving r since z_{x} = z_{y} = 0. Having solved for p and q, equation (10)-(i) then gives us r. Moreover, if u, p, q satisfy the boundary conditions then

6.2.5 so does r. Summarizing, if the boundary conditions are in general position then Thom’s arguments may be used to establish (5) in the case s_{0} = s_{1} = s_{2}, s_{3} = 0.

We now point out that the arguments continue to apply – and indeed are “easier”

– if s_{3} ≠ 0. In this case we may decompose ℝ^{s} = ℝ^{s_{3}} ⊕ ℝ^{s−s_{3}} (s = s_{0} = s_{1} = s_{2}) and correspondingly write v ∈ ℝ^{s} as v = (v′, v′′). The equation (7) is then replaced by

r′′ = F (x, y, z, u, p, q, r′),

and the argument proceeds as before. It is “easier” in the sense not only u but also r′ may be prescribed arbitrarily, subject to open conditions in the C^{1}-topology plus the boundary conditions.

In summary, then, (i) above might be provable by arguments of the type used in [10]. It remains to explain how the absence of real Cartan characteristics allows us to achieve general position for the boundary conditions.

We recall from Chapter V in [2] that characteristics arise when the rank of polar equations of integral elements falls below the maximum. To say that ℐ_{0} has no real Cartan characteristics in the range 0 ≦ q < n − ℓ therefore means that in this range the polar equations have constant rank. If E^{q} is an integral element of dimension q ≧ 1 this rank is s_{1} + ⋯ + s_{q} (the case q = 0 is trivial, and all E^{0} are non-characteristic).

As examples, we note that this condition is satisfied for the contact system on J^{k}(ℝ^{n}, ℝ^{s}), and indeed for any system whose symmetry group acts transitively on the q-dimensional integral elements for 0 ≦ q < n − ℓ. The condition is clearly generic and open. On the other hand, for the exterior differential system arising from the ¯∂-operator in m complex variables, we have n = 2m and ℓ = m; moreover, if m ≧ 3 there are real Cartan characteristics in dimensions 2 ≦ q < m even though the system itself is elliptic.

The way in which the non-characteristic condition arises is this: Referring to the special case s = s_{0} = s_{1} = s_{2}, s_{3} = 0 discussed above, suppose we have f: ∂□^{2} → U ⊂ X_{1}. Then we wish to deform f in a neighborhood of a point p ∈ ∂□ in order to get it in general position at p. The way to do this is to construct a little piece of integral surface near p

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6.2.6 and replace f by f̃ given on the other side of the piece of integral surface. Now since we are in the range 0 ≦ q < n − ℓ constructing the piece of integral surface involves solving finite, not differential equations^{2}, as we saw above. Infinitesimally, constructing the piece of integral surface amounts to solving the polar equations

Θ(ξ, v) = 0, Θ ∈ ℐ_{1}^{2}

where ξ is the tangent to f(∂□) at f(p) and v ∈ T_{f(p)}(X_{1}). So long as these equations do not have jumping rank, we may smoothly do this. In terms of the coordinates used above, the absence of real characteristics means that for any (not just a generic) choice of coordinates (x, y, z) in ℝ^{3} we may solve for r as in (7).

In concluding this discussion, we mention that we are not saying the presence of real Cartan characteristics necessarily provides an obstruction to the validity of (4). What we are saying is that such characteristics provide singularity issues that go beyond those that arose in Thom’s paper, and which clearly merit further investigation.

6.2.7 6.3 Moment Conditions

2What is going on here is that we are discussing the classical strip conditions used to prescribe the boundary values of a PDE in terms of certain number of arbitrary functions and their derivatives. Since we are in the range 0 ≦ q < n − ℓ there are no conditions imposed on the derivatives of these functions, in the sense that the number of equations is insufficient to impose such conditions provided that we are in general position.

We retain the notations of the preceeding section, and shall work locally. In building up integral manifolds the critical dimension is n − ℓ, for it is here that the transition is made from “strip conditions” to what might be called “honest differential equations”.^{1} A very natural question to consider is the Dirichlet problem in the critical dimension: Given an integral manifold

f: S^{n−ℓ} → X,

what are the conditions that it extend to an integral manifold

F: B^{n−ℓ+1} → X.

Here, S^{n−ℓ} and B^{n−ℓ+1} are the standard sphere and ball with ∂B^{n−ℓ+1} = S^{n−ℓ}; since we are working locally we may assume that a topological extension F exists. The “moment conditions” are the homological obstruction to the Dirichlet problem. More formally,

Definition: Let K be the kernel of the natural mapping

H_{n−ℓ,ℐ}(X) → H_{n−ℓ}(X).

Then the moment conditions are

K^{∗} = Hom(K, ℝ).

There is an obvious interesting mapping

(1)  μ: 𝒞 → K^{∗}

from conservation laws to moment conditions, and in this section we will discuss a number of examples of μ. Roughly speaking, it seems that μ should be injective and will sometimes,

6.3.1 but not always, be an isomorphism.^{2} In these examples, which are drawn from those in section 1, we will work locally so that H_{n−ℓ}(X) = 0 and (1) reduces to

1Strip conditions for q-dimensional intregral manifolds in the range 0 ≦ q ≦ n−ℓ are parametrized by s = s_{q} arbitrary functions of q-variables (arbitrary means subject to open conditions in the C^{1}-topology).

In contrast, the integral manifolds of dimension n − ℓ + 1 are given by s_{n−ℓ+1} < s arbitrtary functions of n − ℓ + 1 variables; the remaining s − s_{n−ℓ+1} functions will be subject to non-trivial closed conditions given by the imposition of a PDE system.

(2)  μ: 𝒞 → Hom(H_{n−ℓ,ℐ}(X), ℝ).

We will also follow the notations of that section, and will therefore replace X, ℐ and 𝒞 by X_{0}, ℐ_{0} and 𝒞_{0} as we shall be working with the non-prolonged systems.

Example 1 (The contact system): In this case dim X_{0} = 2n + 1 and ℐ_{0} is generated by

θ = du − ∑_{i=1}^{n} p_{i}dx^{i}.

Also, since ℓ = 0 we should think of H_{n,ℐ_{0}}(X_{0}) as “compactly supported homology”. Thus, “cycles” are given by integral manifolds x → (x, u(x), Du(x)) where u is compactly supported, and no two cycles are homologous since there are no (n+1)-dimensional integral manifolds. The space of conservation laws is given by functionals on integral manifolds, and (2) amounts to the mapping

λ → ∫ λ(x, u(x), Du(x))dx

where u(x) is compactly supported. To prove that μ is an isomorphism means to show that:

(i)  ∫ λ(x, u(x), Du(x))dx = 0

for all compactly supported u ⇒ the class of λ(x, u, p)dx is zero in H̄n 0 – i.e., λdx is a divergence, and

(ii)  ∫ λ(x, u(x), Du(x))dx = ∫ λ(x, ũ(x), Dũ(x))dx

for all λ ⇒ u = ũ.

Both of these are elementary to prove.

6.3.2 Example 2 (the symplectic system): In this case dim X_{0} = 2n, n ≧ 2 and ℐ_{0} is generated by

2We have not proved that it ever fails to be an isomorphism; this seems to require additional foundational work.

Θ = ∑ dp_{i} ∧ dx^{i}.

Also, from section 1 we recall that n − ℓ = 1 and that 𝒞_{0} is one dimensional and generated by any 1-form θ satisfying

dθ = Θ.

The space H_{1,ℐ_{0}}(X_{0}) is given by closed curves modulo the equivalence relation that γ ∼ 0 in case γ = ∂Γ where Γ is a piecewise-C^{1} 2-chain whose simplices are Lagrangian surfaces. We will prove two special cases towards showing that (2) is an isomorphism.

To do this we first consider the case where γ is the image of a mapping

f: S^{1} → X_{0}

where S^{1} is the unit circle in ℝ^{2}. We will show that f extends to a mapping

F: B^{2} → X_{0}

satisfying

F^{∗}Θ = 0

if, and only if, the moment condition

∫_{S^{1}} f^{∗}θ = 0

is satisfied.

Choose any base point P_{0} ∈ S^{1} and set

u(P) = ∫_{P_{0}}^{P} f^{∗}θ.

This defines a function u on S^{1} and we have a diagram

B^{2}  Y
∪ _{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{f̃} ↓^{π}
S^{1} ⟶_{f} X_{0}

6.3.3 where Y = X_{0} × ℝ and f̃ = (f, u) (cf. section 1). By construction, f̃^{∗}(du − θ) = 0 and we may then extend f̃ to a mapping

F̃: B^{2} → Y

satisfying

F̃^{∗}(du − θ) = 0

by the filling in problem for contact systems (cf. Thom [10]). For

F = π ∘ F̃

we have

F^{∗}Θ = −F̃^{∗}(d(du − θ))

= 0

as required.

A slight extension of this argument deals with two mappings

f_{i}: S^{1} → X_{0}, i = 0, 1

which satisfy

(3)  ∫_{S^{1}} f_{0}^{∗}θ = ∫_{S^{1}} f_{1}^{∗}θ

We then want to fill in the cylinder; i.e., to find a mapping

F: S^{1} × I → X_{0}

such that F |S^{1} × {0} = f_{0}, F |S^{1} × {1} = f_{1} and

F^{∗}Θ = 0.

6.3.4 As above, we consider Ỹ = S^{1} × X_{0} and the diagram

S^{1} × I  Ỹ
∪  _{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{f̃}  ↓^{π}
∂(S^{1} × I) ⟶_{f} X_{0}

where f is given by f_{0}, f_{1} and the lifting f̃ exists by virtue of (3) (the u-coordinate is now periodic with the same period ∫_{S^{1}} f_{i}^{∗}θ). By construction, f̃^{∗}(du − θ) = 0 and so again by Thom’s theorem we may extend F̃ to

F̃: S^{1} × I → Ỹ

satisfying F̃^{∗}(du − θ) = 0.

The argument when γ is an arbitrary 1-cycle is more complicated and will be omitted.

Example 3 (CR system): This is the example from section 1 where X_{0} is a complex manifold and ℐ_{0} is generated by the 2-forms α + ᾱ, α ∈ Ω^{2,0}(X_{0}). Since we are working locally we may assume that X_{0} is a contractible, Stein open set in ℂ^{m}, m ≧ 2. Then n = 2, ℓ = 1 and the moment conditions are the linear functionals on H_{1,ℐ_{0}}(X_{0}). On the other hand, as we saw in section 1 the conservation laws are given by

𝒞_{0} ≅ {holomorphic 1-forms ψ/d (holomorphic functions)}

and we are interested in the question of whether the mapping

μ_{0}: 𝒞_{0} → Hom(H_{1,ℐ_{0}}(X_{0}), ℝ)

is an isomorphism. As in the symplectic example just considered, the crucial result is the following theorem of Harvey-Lawson [6]: Given a mapping

f: S^{1} → X_{0}

satisfying

∫_{S^{1}} f^{∗}ψ = 0

6.3.5 for all ψ ∈ 𝒞_{0}, then f extends to a holomorhic mapping of the disc.

As discussed in [6], this result may be extended to show that μ_{0} in (2) is an isomorphism.

Example 4 (Associative geometry): This is the exterior differential system given by the final example in section 1. In this case n = 3, ℓ = 1 and so the crucial homology group is H_{2,ℐ_{0}}(X_{0}). As we saw in the earlier discussion of this example,

dim 𝒞_{0} = 21

and we shall see that this strongly suggests (but does not quite prove) that

μ_{0}: 𝒞_{0} → Hom(H_{2,ℐ_{0}}(X_{0}), ℝ)

fails to be surjective.^{3}

Let E^{3} ⊂ ℝ^{1} be an associated 3-plane with B^{3} ⊂ E^{3} the unit ball having boundary S^{2} = ∂B^{3}. We seek the equations on sections of the normal bundle to B^{3} in ℝ^{7}, T (ℝ^{7})/T (B^{3}) ≅ (ℝ^{7}/E^{3}) × B^{3}, that express the conditions that the corresponding infinitesimal variation of B^{3} remain an integral manifold of ℐ_{0}. In fact, letting F denote the trivial bundle with fiber ℝ^{7}/E^{3} ≅ ℝ^{4}, the linear equations for variation of B^{3} as an integral manifold of ℐ_{0} are just the kernel of the Dirac operator

(4)  D: F → F

(cf. Harvey-Lawson [7]).

Now in general if we have a linear elliptic operator, for convenience still denoted by (4), over the n-ball B ⊂ ℝ^{n}, then for solutions u of D and v of the adjoint operator D^{∗}

∫_{∂B} (u, v) = ∫_{B} (Du, v) + ∫_{B} (u, D^{∗}v)
= 0.

6.3.6 In other words, the boundary values of solutions to the adjoint equation D^{∗}v = 0 give an infinite number of independent moment conditions on boundary values of solutions to the equation Du = 0.^{4}

3This argument only depends on the fact that dim 𝒞_{0} < ∞.

Applying this to variations of B^{3} as an integral manifold of ℐ_{0}, it follows that there are an infinite number of independent conditions imposed on a small perturbation S′ of S^{2} that S′ fill in to an integral manifold. On the other hand, the conservation laws of the system ℐ_{0} provide only a finite number of moment conditions.

There are two ways in which this discussion is incomplete and therefore fails to provide an example where (2) is not an isomorphism. The first is that we have only discussed the moment conditions to fill in S′ with a C^{1} integral manifold. Conceivably, to realize S′ as the boundary of a piecewise-C^{1} chain whose individual simplices are integral manifolds of ℐ_{0} might provide enough additional flexibility to enable (2) to be an isomorphism. Because of regularity for elliptic equations this seems to us unlikely.

The other possibility is that when we pass to the infinite prolongation of ℐ_{0} then H̄2 0 increases to give an infinite dimensional H̄^{2} which accounts for all the moment conditions. What would probably have to happen is that the infinite number of conservation laws for the linearized exterior differential system (4) somehow survive to give conservation laws for ℐ provided that we allow higher order jet dependence. This also seems to us unlikely.

6.3.7

4It is a general fact that for a linear PDE, solutions to the adjoint equation give conservation laws for the original equation. This is the basis for the observation above.

References

[1] I. Anderson, The variational bicomplex, Notes from Dept. of Mathematics at Utah State University (1990).

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[3] R. Bryant, Metrics with exceptional holonomy, Annals of Math. 126 (1987), 525–576.

[4] L. Dickey, On exactness of variational bicomplex, preprint from the Univ. of Oklahoma (1991).

[5] I. M. Gelfand and L. Dikii, Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of Korteweg-de Vries equations, Uspehi Math. Nauk 30 (5) (1975), 67–100.

[6] R. Harvey and H. B. Lawson, On boundaries of complex analytic varieties, I, Annals of Math. 102 (1975), 223–290.

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[9] P. Olver, Applications of Lie groups to differential equations, Springer-Verlag (1986).

[10] R. Thom, Remarques sur les problèmes comportant des inéqualities différentielles globales, Bull. Soc. Math. France 87 (1959), 455–461.

[11] T. Tsujishita, On variational bicomplexes associated to differential equations, Osaka J. of Math. 19 (1982), 311–363.

[12] T. Tsujishita, Homological method of computing invariants of systems of differential equations, Preprint available from Dept. of Mathematics, Osaka Univ. (1991).

[13] A. M. Vinogradov, The 𝒞-spectral sequence, Lagrangian formalism and conservation laws I, II, J. Math. Anal. Appl. 100 (1–129), 1984.

[14] A. M. Vinogradov, Local symmetries and conservation laws, Acta Applicancle Mathematicae 2 (21–78), 1984.

[15] A. M. Vinogradov, Geometry of nonlinear differential equations, J. Soviet Math. 17 (1981), 1624–1649.

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