Moduli stack of vector bundles

<h2 id="definition">Definition</h2>
For the base category, let C be the category of schemes of finite type over a fixed field k. Then 
 
 
 
 
 Vect
 
 n
 
 
 
 
 {\displaystyle \operatorname {Vect} _{n}}
 
 is the category where

<ol><li>an object is a pair 
 
 
 
 (
 U
 ,
 E
 )
 
 
 {\displaystyle (U,E)}
 
 of a scheme U in C and a rank-n vector bundle E over U</li>
<li>a morphism 
 
 
 
 (
 U
 ,
 E
 )
 →
 (
 V
 ,
 F
 )
 
 
 {\displaystyle (U,E)\to (V,F)}
 
 consists of 
 
 
 
 f
 :
 U
 →
 V
 
 
 {\displaystyle f:U\to V}
 
 in C and a bundle-isomorphism 
 
 
 
 
 f
 
 ∗
 
 
 F
 
 
 →
 ∼
 
 
 E
 
 
 {\displaystyle f^{*}F{\overset {\sim }{\to }}E}
 
.</li></ol>
Let 
 
 
 
 p
 :
 
 Vect
 
 n
 
 
 →
 C
 
 
 {\displaystyle p:\operatorname {Vect} _{n}\to C}
 
 be the forgetful functor. Via p, 
 
 
 
 
 Vect
 
 n
 
 
 
 
 {\displaystyle \operatorname {Vect} _{n}}
 
 is a prestack over C. That it is a stack over C is precisely the statement "vector bundles have the <a href="/facts/Descent_(mathematics)/YTMNz2vH">descent</a> property". Note that each fiber 
 
 
 
 
 Vect
 
 n
 
 
 ⁡
 (
 U
 )
 =
 
 p
 
 −
 1
 
 
 (
 U
 )
 
 
 {\displaystyle \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).

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Classifying_stack/lwme8cqw">classifying stack</a></li>
<li><a href="/facts/Moduli_stack_of_principal_bundles/tjRzzo5K">moduli stack of principal bundles</a></li></ul>

<ul><li>Behrend, Kai (2002). "Localization and Gromov-Witten Invariants". In de Bartolomeis; Dubrovin; Reina (eds.). Quantum Cohomology. Lecture Notes in Mathematics. Vol. 1776. Berlin: Springer. pp. 3–38. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-540-45617-9_2">10.1007/978-3-540-45617-9_2</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-43121-3.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">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. Vol. 1776. Berlin: Springer. pp. 3–38. doi:10.1007/978-3-540-45617-9_2. ISBN 978-3-540-43121-3. <a href="https://doi.org/10.1007%2F978-3-540-45617-9_2" target="_blank">https://doi.org/10.1007%2F978-3-540-45617-9_2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Moduli stack of vector bundles open-in-new

Moduli stack of vector bundles