Moduli stack of vector bundles

Template:Short description In algebraic geometry, the moduli stack of rank-n vector bundles Vect_n is the stack parametrizing vector bundles (or locally free sheaves) of rank n over some reasonable spaces.

It is a smooth algebraic stack of the negative dimension $- n^{2}$ .^[1] Moreover, viewing a rank-n vector bundle as a principal $G L_{n}$ -bundle, Vect_n is isomorphic to the classifying stack $B G L_{n} = [pt / G L_{n}] .$

Definition

For the base category, let C be the category of schemes of finite type over a fixed field k. Then ${Vect}_{n}$ is the category where

an object is a pair $(U, E)$ of a scheme U in C and a rank-n vector bundle E over U
a morphism $(U, E) \to (V, F)$ consists of $f : U \to V$ in C and a bundle-isomorphism $f^{*} F \tilde{\to} E$ .

Let $p : {Vect}_{n} \to C$ be the forgetful functor. Via p, ${Vect}_{n}$ 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 ${Vect}_{n} (U) = p^{- 1} (U)$ over U is the category of rank-n vector bundles over U where every morphism is an isomorphism (i.e., each fiber of p is a groupoid).