Sheaf on an algebraic stack

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack ${\displaystyle {\mathfrak {X}}}$ 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 ${\displaystyle \xi }$ in ${\displaystyle {\mathfrak {X}}(S)}$, a quasi-coherent sheaf ${\displaystyle F_{\xi }}$ on S together with maps implementing the compatibility conditions among ${\displaystyle F_{\xi }}$'s.

For a Deligne–Mumford stack, there is a simpler description in terms of a presentation ${\displaystyle U\to {\mathfrak {X}}}$: a quasi-coherent sheaf on ${\displaystyle {\mathfrak {X}}}$ 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.

Definition

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

Let ${\displaystyle {\mathfrak {X}}}$ 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 ${\displaystyle {\mathfrak {X}}}$ is the data consisting of:

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

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

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

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

(cf. equivariant sheaf.)

Examples

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

ℓ-adic formalism

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

See also

• 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)