Moduli stack of vector bundles

Short description: Concept in algebraic geometry

In algebraic geometry, the moduli stack of rank-n vector bundles Vectn 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 −n2.[1] Moreover, viewing a rank-n vector bundle as a principal GLn-bundle, Vectn is isomorphic to the classifying stack BGLn=[pt/GLn].

## Definition

For the base category, let C be the category of schemes of finite type over a fixed field k. Then Vectn 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)->(V,F) consists of f:U->V in C and a bundle-isomorphism f*F->∼E.

Let p:Vectn->C be the forgetful functor. Via p, Vectn 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 Vectn(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 2002, Example 20.2.

* Behrend, Kai (2002). "Localization and Gromov-Witten Invariants". in de Bartolomeis; Dubrovin; Reina. Quantum Cohomology. Lecture Notes in Mathematics. Lecture Notes in Mathematics. 1776. Berlin: Springer. pp. 3—38. https://doi.org/10.1007/978-3-540-45617-9_2.

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