Moduli stack of vector bundles: Difference between revisions

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{{Short description|Concept in algebraic geometry}}

In algebraic geometry, the '''moduli stack of rank -''n'' vector bundles''' Vect_''n'' is the [[stack (mathematics)|stack]] parametrizing [[vector bundle]]s (or [[locally free sheaf|locally free sheaves]]) of rank ''n'' over some reasonable spaces.

It is a [[smooth stack|smooth]] [[algebraic stack|algebraic]] stack of the negative dimension <math>-n^2</math>.<ref>{{harvnb|Behrend|2002|loc=Example 20.2.}}</ref> Moreover, viewing a rank-''n'' vector bundle as a principal <math>GL_n</math>-bundle, Vect_''n'' is isomorphic to the [[classifying stack]] <math>BGL_n = [\text{pt}/GL_n].</math>

== Definition ==

For the base category, let ''C'' be the category of schemes of finite type over a fixed field ''k''. Then <math>\operatorname{Vect}_n</math> is the category where

# an object is a pair <math>(U, E)</math> of a scheme ''U'' in ''C'' and a rank-''n'' vector bundle ''E'' over ''U''

# a morphism <math>(U, E) \to (V, F)</math> consists of <math>f: U \to V</math> in ''C'' and ana [[bundle-isomorphism]] <math>f^* F \overset{\sim}\to E</math>.

Let <math>p: \operatorname{Vect}_n \to C</math> be the forgetful functor. Via ''p'', <math>\operatorname{Vect}_n</math> is a prestack over ''C''. That it is a stack over ''C'' is precisely the statement "vector bundles have the [[descent (mathematics)|descent]] property". Note that each fiber <math>\operatorname{Vect}_n(U) = p^{-1}(U)</math> 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]]

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== References ==

{{reflist}}

*Kai Behrend; Localization and Gromov-Witten invariants; Lecture 1

*{{cite book |last=Behrend |first=Kai |year=2002 |chapter=Localization and Gromov-Witten Invariants |title=Quantum Cohomology. Lecture Notes in Mathematics |editor1-last=de Bartolomeis |editor2-last=Dubrovin |editor3-last=Reina |volume=1776 |publisher=Springer |___location=Berlin |doi=10.1007/978-3-540-45617-9_2 |pages=3–38|isbn=978-3-540-43121-3 }}

[[Category:Algebraic geometry]]

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