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 ${\displaystyle -n^{2}}$.[1] Moreover, viewing a rank-n vector bundle as a principal ${\displaystyle GL_{n}}$-bundle, Vect_n is isomorphic to the classifying stack ${\displaystyle BGL_{n}=[{\text{pt}}/GL_{n}].}$

Definition

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

1. Behrend, Example 20.2. harvnb error: no target: CITEREFBehrend (help)

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