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 $-n^2$. Moreover, viewing a rank-''n'' vector bundle as a principal $GL_n$-bundle, Vect_''n'' is isomorphic to the classifying stack BGL_n = text/GL_n

Definition

For the base category, let ''C'' be the category of schemes of finite type over a fixed field ''k''. Then $\operatorname_n$ is the category where
# an object is a pair $(U, E)$ of a scheme ''U'' in ''C'' and a rank-''n'' vector bundle ''E'' over ''U''
# a morphism $(U, E) \to (V, F)$ consists of $f: U \to V$ in ''C'' and a bundle-isomorphism $f^* F \overset\to E$.

Let $p: \operatorname_n \to C$ be the forgetful functor. Via ''p'', $\operatorname_n$ 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 $\operatorname_n(U) = p^(U)$ 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

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{{algebraic-geometry-stub

Algebraic geometry