Sheaf on an algebraic stack

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack ${\displaystyle {\mathfrak {X}}}${\displaystyle {\mathfrak {X}}} is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is a data that consists of, for each a scheme S in the base category and ${\displaystyle \xi }${\displaystyle \xi } in ${\displaystyle {\mathfrak {X}}(S)}${\displaystyle {\mathfrak {X}}(S)}, a quasi-coherent sheaf ${\displaystyle F_{\xi }}${\displaystyle F_{\xi }} on S together with maps implementing the compatibility conditions among ${\displaystyle F_{\xi }}${\displaystyle F_{\xi }}'s.

For a Deligne–Mumford stack, there is a simpler description in terms of a presentation ${\displaystyle U\to {\mathfrak {X}}}${\displaystyle U\to {\mathfrak {X}}}: a quasi-coherent sheaf on ${\displaystyle {\mathfrak {X}}}${\displaystyle {\mathfrak {X}}} is one obtained by descending a quasi-coherent sheaf on U.^[1] A quasi-coherent sheaf on a Deligne–Mumford stack generalizes an orbibundle (in a sense).

Constructible sheaves (e.g., as l-adic sheaves) can also be defined on an algebraic stack and they appear as coefficients of cohomology of a stack.

The following definition is (Arbarello, Cornalba & Griffiths 2011, Ch. XIII., Definition 2.1.)

Let ${\displaystyle {\mathfrak {X}}}${\displaystyle {\mathfrak {X}}} be a category fibered in groupoids over the category of schemes of finite type over a field with the structure functor p. Then a quasi-coherent sheaf on ${\displaystyle {\mathfrak {X}}}${\displaystyle {\mathfrak {X}}} is the data consisting of:

1. for each object ${\displaystyle \xi }${\displaystyle \xi }, a quasi-coherent sheaf ${\displaystyle F_{\xi }}${\displaystyle F_{\xi }} on the scheme ${\displaystyle p(\xi )}${\displaystyle p(\xi )},
2. for each morphism ${\displaystyle H:\xi \to \eta }${\displaystyle H:\xi \to \eta } in ${\displaystyle {\mathfrak {X}}}${\displaystyle {\mathfrak {X}}} and ${\displaystyle h=p(H):p(\xi )\to p(\eta )}${\displaystyle h=p(H):p(\xi )\to p(\eta )} in the base category, an isomorphism

   ${\displaystyle \rho _{H}:h^{*}(F_{\eta }){\overset {\simeq }{\to }}F_{\xi }}${\displaystyle \rho _{H}:h^{*}(F_{\eta }){\overset {\simeq }{\to }}F_{\xi }}

satisfying the cocycle condition: for each pair ${\displaystyle H_{1}:\xi _{1}\to \xi _{2},H_{2}:\xi _{2}\to \xi _{3}}${\displaystyle H_{1}:\xi _{1}\to \xi _{2},H_{2}:\xi _{2}\to \xi _{3}},

${\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {h_{1}^{*}(\rho _{H_{2}})}{\to }}h_{1}^{*}F_{\xi _{2}}{\overset {\rho _{H_{1}}}{\to }}F_{\xi _{1}}}${\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {h_{1}^{*}(\rho _{H_{2}})}{\to }}h_{1}^{*}F_{\xi _{2}}{\overset {\rho _{H_{1}}}{\to }}F_{\xi _{1}}} equals ${\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {\sim }{=}}(h_{2}\circ h_{1})^{*}F_{\xi _{3}}{\overset {\rho _{H_{2}\circ H_{1}}}{\to }}F_{\xi _{1}}}${\displaystyle h_{1}^{*}h_{2}^{*}F_{\xi _{3}}{\overset {\sim }{=}}(h_{2}\circ h_{1})^{*}F_{\xi _{3}}{\overset {\rho _{H_{2}\circ H_{1}}}{\to }}F_{\xi _{1}}}.

(cf. equivariant sheaf.)

• The Hodge bundle on the moduli stack of algebraic curves of fixed genus.

The l-adic formalism (theory of l-adic sheaves) extends to algebraic stacks.

• Hopf algebroid - encodes the data of quasi-coherent sheaves on a prestack presentable as a groupoid internal to affine schemes (or projective schemes using graded Hopf algebroids)

• https://mathoverflow.net/questions/69035/the-category-of-l-adic-sheaves
• http://math.stanford.edu/~conrad/Weil2seminar/Notes/L16.pdf Adic Formalism, Part 2 Brian Lawrence March 1, 2017

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