Moduli stack of vector bundles explained
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
. 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
- 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
p:\operatorname{Vect}n\toC
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
\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
References
- Book: Behrend, Kai . 2002 . Localization and Gromov-Witten Invariants . Quantum Cohomology. Lecture Notes in Mathematics . de Bartolomeis . Dubrovin . Reina . Lecture Notes in Mathematics . 1776 . Springer . Berlin . 10.1007/978-3-540-45617-9_2 . 3–38.
