Grassmann sheaves and the classification of vector sheaves

TL;DR

This paper develops a framework using Grassmann sheaves to classify vector sheaves over paracompact spaces, establishing a correspondence between vector sheaves and sections of a universal Grassmann sheaf.

Contribution

It introduces the construction of Grassmann sheaves for algebraic sheaves and proves their role in classifying vector sheaves over paracompact spaces.

Findings

Construction of Grassmann sheaves G_A(k,n)

Every vector sheaf is a subsheaf of A^{ abla}

Vector sheaves of rank n are classified by sections of G_A(n)

Abstract

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).