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).

💡 Research Summary

The paper develops a sheaf‑theoretic analogue of the classical Grassmannian classification of vector bundles, working over a sheaf of commutative unital algebras ( \mathcal{A} ) on a topological space ( X ). The authors begin by fixing ( \mathcal{A} ) and the free sheaf ( \mathcal{A}^n ). For each integer ( k \le n ) they define a presheaf that assigns to an open set ( U \subset X ) the set of locally free ( \mathcal{A}|U )-subsheaves of rank ( k ) inside ( \mathcal{A}^n|U ). After sheafifying this presheaf they obtain the (k)-th Grassmann sheaf ( \mathcal{G}{\mathcal{A}}(k,n) ). By construction, a global section of ( \mathcal{G}{\mathcal{A}}(k,n) ) is precisely a rank‑(k) locally free subsheaf of ( \mathcal{A}^n ) over the whole space.

The second major result concerns the embedding of arbitrary vector sheaves into an infinite direct sum of copies of ( \mathcal{A} ). Assuming that ( X ) is paracompact, the authors use a locally finite open cover together with a continuous partition of unity subordinate to that cover. On each member ( U_i ) of the cover the given vector sheaf ( \mathcal{E} ) is locally free, hence isomorphic to ( \mathcal{A}^{r_i}|_{U_i} ). The partition of unity provides coefficients that allow the local isomorphisms to be glued into a single monomorphism \