Moduli stack of vector bundles
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 .[1] Moreover, viewing a rank-n vector bundle as a principal -bundle, Vectn 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
- an object is a pair of a scheme U in C and a rank-n vector bundle E over U
- 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
References
- ^ Behrend 2002, Example 20.2.
- Behrend, Kai (2002). "Localization and Gromov-Witten Invariants". In de Bartolomeis; Dubrovin; Reina (eds.). Quantum Cohomology. Lecture Notes in Mathematics. Lecture Notes in Mathematics. Vol. 1776. Berlin: Springer. pp. 3–38.