Grassmann sheaves and the classification of vector sheaves

Given a sheaf of unital commutative and associative algebras A, first we construct the k-th Grassmann sheaf G_A(k,n) of A^n whose sections induce vector subsheaves of A^n of rank k. Next we show that every vector sheaf over a paracompact space is a subsheaf of A^{\infty}. Finally, applying the preceding results to the universal Grassmann sheaf G_A(n), we prove that vector sheaves of rank n over a paracompact space are classified by the global sections of G_A(n).


Introduction
Let A be a sheaf of unital commutative and associative algebras over the ring R or C. A vector sheaf E is a locally free A-module. For instance, the sections of a vector bundle provide such a sheaf. However, a vector sheaf is not necessarily free, as is the case of the sections of a non trivial vector bundle.
Recently, vector sheaves gained a particular interest because they serve as the platform to abstract the classical geometry of vector bundles and their connections within a non smooth framework. This point of view has already been developed in [7] (see also [8] for applications to physics, and [10] for the reduction of the geometry of vector sheaves to the general setting of principal sheaves).
A fundamental result of the classical theory is the homotopy classification of vector bundles (of rank, say, n) over a fixed base. The construction of the classifying space, and the subsequent classification, are based on the Grassmann manifold (or variety) G k (R n ) of k-dimensional subspaces of R n . In this respect we refer, e.g., to [5] and [6]. However, considering vector sheaves, we see that a homotopy classification is not possible, since the pull backs of a vector sheaf by homotopic maps need not be isomorphic, even in the trivial case of the free A-module A, as we prove in Section 1. Consequently, any attempt to classify vector sheaves (over a fixed space X) should not involve pull-backs and homotopy.
In this paper we develop a classification scheme based on a sort of universal Grassmann sheaf. More explicitly, for fixed k ≤ n ∈ N, in Section 2 we construct -in two equivalent ways-a sheaf G A (k, n), legitimately called the k-th Grassmann sheaf of A n , whose sections coincide (up to isomorphism) with vector subsheaves of A n of rank k (Proposition 2.3). Then, inducing in Section 3 the vector sheaf A ∞ , we show that every vector sheaf over a paracompact space is a subsheaf of A ∞ (Theorem 3.1). A direct application of the previous ideas leads us to the construction of the universal Grassmann sheaf G A (n) of rank n. The main result here (Theorem 3.5) asserts that arbitrary vector sheaves of rank n, over a paracompact base space, coincide -up to isomorphism-with the sections of G A (n).

Vector sheaves and homotopy
For the general theory of sheaves we refer to standard sources such as [1], [2], [4], and [9]. In what follows we recall a few definitions in order to fix the notations and terminology of the present paper.
Throughout the paper A denotes a fixed sheaf of unital commutative and associative K-algebras (K = R, C) over a topological space X. An Amodule E ≡ (E, π, X) is a sheaf whose stalks E x are A x -modules so that the respective operations of addition and scalar multiplication E × X E −→ E and A × X E −→ E are continuous. In particular, a vector sheaf of rank n is an A-module E, locally isomorphic to A n . This means there is an open covering U = {U α }, α ∈ I, of X and A| Uα -isomorphisms The category of vector sheaves of rank n over X is denoted by V n (X). More details, examples and applications of vector sheaves can be found in [7], [8], and [10].
As already mentioned in the Introduction, we shall show, by a concrete counterexample, that homotopic maps do not yield isomorphic pull-backs, even in the simplest case of the free A-module A. In fact, we consider two non-isomorphic algebras A 0 and A 1 and a morphism of algebras ρ : A 0 → A 1 . Given now a topological space X and a fixed point It is not difficult to show that A(U ), ρ U V is a presheaf whose sheafification is a sheaf of algebras, denoted by A ≡ (A, π, X). It is clear that On the other hand, if there is an which is not isomorphic to A x 0 . Hence, a vector sheaf, even a free one, need not have locally isomorphic fibres. Let now α : [0, 1] → X be a continuous path with α(0) = x 0 and α(1) = x 1 . For any topological space Y , we define the map Obviously, this is a homotopy between the constant maps f 0 = x 0 : Y → X and f 1 = x 1 : Y → X. As a result, taking the pull-backs of A by the latter, we see that, for every y ∈ Y , that is, we obtain two non-isomorphic stalks, thus proving the claim.

The Grassmann sheaf of rank k in A n
As in Section 1, A is a sheaf of unital commutative and associative Kalgebras over a given topological space X. We denote by T X the topology of X and fix n ∈ N. For k ∈ N with k ≤ n and any U ∈ T X , we define the set denotes the natural restriction, it is clear that the collection is a vector sheaf over U , but not necessarily a free A| U -module.
Definition 2.1. The k-th Grassmann sheaf of A n , denoted by G A (k, n), is defined to be the sheaf generated by the presheaf G A (k, n).
Since G A (k, n) is not complete, it does not coincide with the complete presheaf of (continuous) sections of G A (k, n). We shall describe G A (k, n) via another complete presheaf. As a matter of fact, we consider the presheaf where now Proof. Clearly, for every U ∈ T X , Since V A (k, n) is complete, it is isomorphic with the sheaf of sections of G A (k, n), thus we have the following interpretation of the elements of G A (k, n)(X).

The universal Grassmann sheaf
The preliminary results of the preceding section hold for every base space X. Here we prove that if X is a paracompact space, then any vector sheaf can be interpreted as a section of an appropriate universal Grassmann sheaf.
First we prove a Whitney-type embedding theorem. To this end, for every sheaf of algebras A, we consider the presheaf where A i = A, for every i ∈ N, with the obvious restrictions. This presheaf generates the infinite fibre product which is a free A-module. Then, we obtain: Theorem 3.1. Let X be a paracompact space. Then every vector sheaf E of finite rank over X is a subsheaf of A ∞ .
Proof. Let E be a vector sheaf of rank, say, k. Since X is paracompact, a reasoning similar to that of [5,Proposition 5.4] proves that E is free over a countable open covering {U i } i∈N of X. Let ψ i : Therefore, α i ψ i is an A-module morphism, whose restriction to the interior of supp α i ⊆ U i is an isomorphism. We consider the fibre product i∈N (A k ) i , where (A k ) i ≡ A k , for every i ∈ N, and we denote by the corresponding projections. The universal property of the product ensures the existence of a unique A-morphism Then ψ is a monomorphism. In fact, let 0 = u ∈ E x with ψ x (u) = 0. There where A i = A, for every i ∈ N, the assertion is proven.
We shall show that a further restriction on the topology of X leads to an embedding of E into a smaller sheaf. To this end, assume that E is a vector sheaf of rank k, which is free over a finite open covering {U i } 1≤i≤n of X. Let ψ i : E| U i → A k | U i be the respective A-module isomorphisms, and {α i : X → R} 1≤i≤n a subordinate partition of unity. Considering the maps α i ψ i : E → A k , as before, we obtain the sheaf morphism f : E −→ A kn : u → α 1 (π(u)) · ψ 1 (u), . . . , α n (π(u)) · ψ n (u) which embeds E into the free A-module A kn . Therefore we have proved the following: Proposition 3.2. Over a compact space X, every vector sheaf of finite rank is a subsheaf of a free A-module of finite rank.
Clearly, every free A-module A n is a submodule of the free A-module i∈N A i , with A i = A, for every i ∈ N. Thus we obtain: Corollary 3.3. Over a compact space X, every vector sheaf of finite rank is a subsheaf of the free A-module i∈N A i .
We are now in a position to repeat the constructions of Section 2 in a more general way. For every n ∈ N, we define the set G A (n)(U ) := S subsheaf of A ∞ | U : S ∼ = A n | U .

Then the collection
where ρ U V denotes the obvious restriction, is a non-complete monopresheaf. and λ U V are the natural restrictions. As a result, adapting the proof of Proposition 2.3 to the present situation, we obtain the main result of this work, namely the following classification of vector sheaves: Theorem 3.5. If X is a paracompact space, then the vector sheaves of rank n (over X) coincide -up to isomorphism-with the global sections of the universal Grassmann sheaf G A (n).