Moduli stack of vector bundles


In algebraic geometry, the moduli stack of rank-n vector bundles Vectn is the stack parametrizing vector bundles of rank n over some reasonable spaces.
It is a smooth algebraic stack of the negative dimension. Moreover, viewing a rank-n vector bundle as a principal -bundle, Vectn is isomorphic to the classifying stack

Definition

For the base category, let C be the category of schemes of finite type over a fixed field k. Then is the category where
  1. an object is a pair of a scheme U in C and a rank-n vector bundle E over U
  2. a morphism consists of in C and a bundle-isomorphism.
Let be the forgetful functor. Via p, is a prestack over C. That it is a stack over C is precisely the statement "vector bundles have the descent property". Note that each fiber over U is the category of rank-n vector bundles over U where every morphism is an isomorphism.