A short note on vector bundles on curves

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.

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