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.

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

Definition

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

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

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