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 the data 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.) harv error: no target: CITEREFArbarelloCornalbaGriffiths2011 (help)

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 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)

Notes

1. Arbarello, Cornalba & Griffiths 2011, Ch. XIII., § 2. harvnb error: no target: CITEREFArbarelloCornalbaGriffiths2011 (help)

References

• Enrico Arbarello, Maurizio Cornalba, and Phillip Griffiths, Geometry of algebraic curves. Vol. II, with a contribution by Joseph Daniel Harris, Grundlehren der Mathematischen Wissenschaften, vol. 268, Springer, Heidelberg, 2011. MR2807457 doi:10.1007/978-1-4757-5323-3
• Behrend, Kai (2003), "Derived l-adic categories for algebraic stacks", Memoirs of the American Mathematical Society, 774
• Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 39, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65761-3, MR 1771927
• Olsson, Martin (2007). "Sheaves on Artin stacks". Journal für die reine und angewandte Mathematik. 603: 55–112. Editorial note: This paper corrects a mistake in Laumon and Moret-Bailly's Champs algébriques.
• Rydh, David (2016). "Approximation of sheaves on algebraic stacks". International Mathematics Research Notices. 2016 (3): 717–737. arXiv:1408.6698.

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