Sheaf on an algebraic stack

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

For a Deligne–Mumford stack, there is a simpler description in terms of a presentation $U\to 𝔛$: a quasi-coherent sheaf on $𝔛$ is one obtained by descending a quasi-coherent sheaf on U. 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 Template:Harv

Let $𝔛$ 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 $𝔛$ is the data consisting of:

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

   ${\rho }_H:h^{*}(F_{\eta })\stackrel{\simeq }{\to }{F}_{\xi }$

satisfying the cocycle condition: for each pair $H_1:{\xi }_1\to {\xi }_2,H_2:{\xi }_2\to {\xi }_3$,

$h_1^{*}{h}_2^{*}{F}_{{\xi }_3}\stackrel{h_1^{*}({\rho }_{H_2})}{\to }{h}_1^{*}{F}_{{\xi }_2}\stackrel{{\rho }_{H_1}}{\to }{F}_{{\xi }_1}$ equals $h_1^{*}{h}_2^{*}{F}_{{\xi }_3}\stackrel{\sim }{=}(h_2\circ h_1{)}^{*}{F}_{{\xi }_3}\stackrel{{\rho }_{H_2\circ H_1}}{\to }{F}_{{\xi }_1}$.

(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)

References

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• Template:Cite journal Editorial note: This paper corrects a mistake in Laumon and Moret-Bailly's Champs algébriques.
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External links

• 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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