Moduli stack of vector bundles

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.

Definition

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

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

Let ${\displaystyle p:\operatorname {Vect} _{n}\to C}$  be the forgetful functor. Via p, ${\displaystyle \operatorname {Vect} _{n}}$  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 ${\displaystyle \operatorname {Vect} _{n}(U)=p^{-1}(U)}$  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).

See also

• classifying stack
• moduli stack of principal bundles

References

• Kai Behrend; Localization and Gromov-Witten invariants; Lecture 1