1. Definition

2. See also

3. References

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 .^[1] Moreover, viewing a rank-n vector bundle as a principal -bundle, Vect_n is isomorphic to the classifying stack

Definition

For the base category, let C be the category of schemes of finite type over a fixed field k. Then is the category where

1. an object is a pair of a scheme U in C and a rank-n vector bundle E over U
2. a morphism consists of in C and a bundle-isomorphism .

Let be the forgetful functor. Via p, 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 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

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