M ay 2 00 9 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 GA(k, n) of An, whose sections induce vector subsheaves of An of rank k. Next we show that every vector sheaf over a paracompact space is a subsheaf of A. Finally, applying the preceding results to the universal Grassmann sheaf GA(n), we prove that vector sheaves of rank n over a paracompact space are classified by the global sections of GA(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 2000 Mathematics Subject Classification. Primary: 18F20. keywords. Vector sheaves, Grassmann sheaves. ∗The authors were partially supported by University of Athens Research Grands 70/4/5639 and 70/4/3410, respectively. 2 M. H. Papatriantafillou –E. Vassiliou Grassmann manifold (or variety) Gk(R ) of k-dimensional subspaces of R. 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 GA(k, n), legitimately called the k-th Grassmann sheaf of A, whose sections coincide (up to isomorphism) with vector subsheaves of A 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 GA(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 GA(n). 1. 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 Ex are Ax-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. This means there is an open covering U = {Uα}, α ∈ I, of X and A|Uα-isomorphisms ψα : E|Uα −→ A |Uα , α ∈ I. The category of vector sheaves of rank n over X is denoted by V(X). More details, examples and applications of vector sheaves can be found in [7], [8], and [10]. Classification of Vector Sheaves 3 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 A0 andA1 and a morphism of algebras ρ : A0 → A1. Given now a topological space X and a fixed point x0 ∈ X, for every open U ⊆ X we set A(U) := { A0, x0 ∈ U, A1, x0 / ∈ U, while, for every open V ⊆ U , ρUV : A(U) → A(V ) denotes the corresponding (restriction) map, defined by