Descent data and twisted forms
A natural way to study vector bundles is to use the notion of transition functions. Given any vector bundle, you trivialise it on an open cover so it becomes a product with a finite dimensional vector space V. You reconstruct it with transition functions which twist the gluing so you do not necessarily get a global product with V. In this video, we show how to generalise this notion of transition functions to that of descent data in a Grothendieck topology. This allows you to study objects which are locally like some object W, where locally is now in some Grothendieck topology. This gives you a twisted form of W. An interesting example we look at is that of Azumaya algebras which are locally a product with a full matrix algebra.