COHOMOLOGY OF THE VARIATIONAL COMPLEX IN THE CLASS OF EXTERIOR FORMS OF FINITE JET ORDER

We obtain the cohomology of the variational complex on the infinite-order jet space of a smooth fiber bundle in the class of exterior forms of finite jet order. In particular, this provides a solution of the global inverse problem of the calculus of variations of finite order on fiber bundles.

problem of the calculus of variations in the class of exterior forms of finite jet order is provided.
2. The differential calculus on J ∞ Y . Smooth manifolds throughout are assumed to be real, finite-dimensional, Hausdorff, second-countable (i.e., paracompact), and connected. We follow the terminology of [4,8], where a sheaf S is a particular topological bundle,S denotes the canonical presheaf of sections of the sheaf S, and Γ (S) is the group of global sections of S.
Recall that the infinite-order jet space J ∞ Y of a smooth fiber bundle Y → X is defined as a projective limit of the inverse system of finite-order jet manifolds J r Y of Y → X. Endowed with the projective limit topology, J ∞ Y is a paracompact Fréchet manifold [12]. A bundle coordinate atlas {U Y ,(x λ ,y i )} of Y → X yields the manifold coordinate atlas of graded differential R-algebras ᏻ * r of exterior forms on finite-order jet manifolds J r Y , where π r r −1 * are the pullback monomorphisms. The direct limit of this direct system is the above mentioned graded differential algebra ᏻ * ∞ of exterior forms on finite-order jet manifolds modulo the pullback identification. By passing to the direct limit of the de Rham complexes of exterior forms on finite-order jet manifolds, the de Rham cohomology of ᏻ * ∞ has been proved to coincide with the de Rham cohomology of the fiber bundle Y [1,3]. However, this is not a way of studying other cohomologies of the algebra ᏻ * ∞ . To solve this problem, we enlarge ᏻ * ∞ as follows. Let O * r be a sheaf of germs of exterior forms on the r -order jet manifold J r Y , and O * r its canonical presheaf. There is the direct system of canonical presheaves where π r r −1 * are the pullback monomorphisms. Its direct limitŌ * ∞ is a presheaf of graded differential R-algebras on J ∞ Y . Let T * ∞ be a sheaf constructed fromŌ * ∞ ,T * ∞ its canonical presheaf, and - * ∞ = Γ (T * ∞ ) the structure algebra of sections of the sheaf T * ∞ . There are R-algebra monomorphismsŌ * ∞ →T * ∞ and ᏻ * ∞ → - * ∞ . The key point is that, since the paracompact space J ∞ Y admits a partition of unity by elements of the ring -0 ∞ [12], the sheaves of -0 ∞ -modules on J ∞ Y are fine and, consequently, acyclic. Therefore, the abstract de Rham theorem on cohomology of a sheaf resolution [8] can be called into play in order to obtain cohomology of the graded differential algebra - * ∞ . Then we prove that ᏻ * ∞ has the same d H -and δ-cohomology as - * ∞ . For short, we agree to call elements of - * ∞ the exterior forms on J ∞ Y . Restricted to a coordinate chart ( ∞ of k-contact and s-horizontal forms, together with the corresponding projections Accordingly, the exterior differential on - * ∞ is split into the sum d = d H + d V of horizontal and vertical differentials such that, (2.9)

The variational bicomplex.
Being nilpotent, the differentials d V and d H provide To complete it to the variational bicomplex, we define the projection R-module endomorphism: Introduced on elements of the presheafŌ * ∞ (cf. [3,5,13]), this endomorphism is induced on the sheaf T * ∞ and its structure algebra - * ∞ . Put Since τ is a projection operator, we have isomorphisms The variational operator on T * ,n ∞ is defined as the morphism δ = τ • d. It is nilpotent, and obeys the relation (3.5) Let R and O * X denote the constant sheaf on J ∞ Y and the sheaf of exterior forms on X, respectively. The operators d V , d H , τ, and δ give the following variational bicomplex of sheaves of differential forms on J ∞ Y : The second row and the last column of this bicomplex form the variational complex The corresponding variational bicomplexes and variational complexes of graded differential algebras - * ∞ and ᏻ * ∞ take place. There are the well-known statements summarized usually as the algebraic Poincaré lemma (cf. [11,13]).
It follows that the variational bicomplex (3.6) and, consequently, the variational complex (3.7) are exact for any smooth bundle Y → X. Moreover, the sheaves T k,m ∞ in this bicomplex are fine, and so are the sheaves E k in accordance with the following lemma. Proof. Although the R-modules E k>1 fail to be -0 ∞ -modules [13], we can use the fact that the sheaves E k>0 are projections τ(T k,n of the sheaves E k . They possess the properties required for E k to be a fine sheaf. Indeed, for each i ∈ I, suppf i ⊂ U i provides a closed set such thath i is zero outside this set, while the sum i∈Ihi is the identity morphism. Thus, the columns and rows of bicomplex (3.6) as well as the variational complex (3.7) are sheaf resolutions, and the abstract de Rham theorem can be applied to them. Here, we restrict our consideration to the variational complex.

Cohomology of - *
∞ . The variational complex (3.7) is a resolution of the constant sheaf R on J ∞ Y . We start from the following lemma.

Lemma 4.1. There is an isomorphism
, the first isomorphism in (4.1) follows from the Vietoris-Begle theorem [4], while the second one results from the familiar de Rham theorem.
Consider the de Rham complex of sheaves on J ∞ Y and the corresponding de Rham complex of their structure algebras Complex (4.2) is exact due to the Poincaré lemma, and is a resolution of the constant sheaf R on J ∞ Y since sheaves T r ∞ are fine. Then, the abstract de Rham theorem and Lemma 4.1 lead to the following.
It follows that every closed form φ ∈ - * ∞ is split into the sum where ϕ is a closed form on the fiber bundle Y . Similarly, from the abstract de Rham theorem and Lemma 4.1, we obtain the following.

Proposition 4.3. There is an isomorphism between d H -and δ-cohomology of the variational complex
and the de Rham cohomology of the fiber bundle Y , namely, This isomorphism recovers the results of [2,12], but notes also the following. Relation where ϕ is a closed m-form on Y . Any δ-closed form σ ∈ -k,n , k ≥ 0, is split into where ϕ is a closed (n + k)-form on Y .
Proof. Let the common symbol D stand for d H and δ. Bearing in mind the decompositions (4.7), (4.8), (4.9), and (4.10), it suffices to show that, if an element φ ∈ ᏻ * ∞ is D-exact in the algebra - * ∞ , then it is so in the algebra ᏻ * ∞ . Lemma 3.1 states that, if Y is a contractible bundle and a D- Moreover, a glance at the homotopy operators for d H and δ (see [11]) shows that the jet order Proof of (i). Let φ ∈ ᏻ * ∞ be a D-exact form on J ∞ Y . The finite exactness on (π ∞ 0 ) −1 (∪U α ) holds since φ = Dϕ α on every (π ∞ 0 ) −1 (U α ) and [ϕ α ] < N([φ]). Proof of (ii). Let φ = Dϕ ∈ ᏻ * ∞ be a D-exact form on J ∞ Y . By assumption, it can be brought into the form Dϕ U on (π ∞ 0 ) −1 (U ) and Dϕ V on (π ∞ 0 ) −1 (V ), where ϕ U and ϕ V are exterior forms of bounded jet order. We consider their difference which, by assumption, can be written as ϕ U − ϕ V = Dσ where σ is also of bounded jet order [σ ] < N (N([φ])). Lemma 5.2 below shows that σ = σ U + σ V where σ U and σ V are exterior forms of bounded jet order on (π ∞ 0 ) −1 (U) and (π ∞ 0 ) −1 (V ), respectively. Then, putting is of bounded jet order [ϕ ] < N (N([φ])). To prove the finite exactness of D on J ∞ Y , it remains to choose an appropriate cover of Y . A smooth manifold Y admits a countable cover {U ξ } by domains U ξ , ξ ∈ N, and its refinement {U ij }, where j ∈ N and i runs through a finite set, such that U ij ∩ U ik = ∅, j ≠ k [7]. Then Y has a finite cover {U i = j U ij }. Since the finite exactness of the operator D takes place over any domain U ξ , it also holds over any member U ij of the refinement {U ij } of {U ξ } and, in accordance with Theorem 5.1 item (i) above, over any member of the finite cover {U i } of Y . Then by virtue of item (ii) above, the finite exactness of D takes place over Y .
Proof. By taking a smooth partition of unity on U ∪ V subordinate to the cover {U,V } and passing to the function with support in V , we get a smooth real function f on U ∪ V which is 0 on a neighborhood of U − V , and 1 on a neighborhood of The exterior form ((π ∞ 0 ) * f )σ is 0 on a neighborhood of (π ∞ 0 ) −1 (U) and, therefore, can be extended by 0 to (π ∞ 0 ) −1 (U ). We denote it by σ U . Accordingly, the exterior form (1 − (π ∞ 0 ) * f )σ has an extension σ V by 0 to (π ∞ 0 ) −1 (V ). Then, σ = σ U +σ V is a desired decomposition because σ U and σ V are of the jet order which does not exceed that of σ .
where ϕ is a closed n-form on Y .
(ii) A finite-order Euler-Lagrange-type operator satisfies the Helmholtz condition δ(Ᏹ) = 0 if and only if where φ is a closed (n + 1)-form on Y .
Note that item (i) in Corollary 6.1 contains the result of [14]. As was mentioned above, the theses of Corollary 6.1 also agree with those of [2], but the proof of Theorem 5.1 does not give a sharp bound on the order of a Lagrangian.

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