A short note on vector bundles on curves

TL;DR

This paper provides a simplified proof of the correspondence between the affine Grassmannian for GL_n and moduli spaces of vector bundles on curves, avoiding complex descent lemmas through an approximation argument.

Contribution

It introduces a more direct proof of the vector bundle correspondence, bypassing the need for an abstract descent lemma by using an approximation approach.

Findings

Simplified proof of the vector bundle and affine Grassmannian correspondence.

Avoidance of complex descent lemmas through approximation methods.

Enhanced understanding of vector bundle moduli on curves.

Abstract

Beauville and Laszlo give an interpretation of the affine Grassmannian for Gl_n over a field k as a moduli space of, loosely speaking, vector bundles over a projective curve together with a trivialization over the complement of a fixed closed point. In order to establish this correspondence, they have to show that descent for vector bundles holds in a situation which is not a classical fpqc-descent situation. They prove this as a consequence of an abstract descent lemma. It turns out, however, that one can avoid this descent lemma by using a simple approximation-argument, which leads to a more direct prove of the above mentioned correspondence.