Sheaf on an algebraic stack

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack ${\mathfrak {X}}$ 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 $\xi$ in ${\mathfrak {X}}(S)$ , 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 $U\to {\mathfrak {X}}$ : a quasi-coherent sheaf on ${\mathfrak {X}}$ 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 ${\mathfrak {X}}$ 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 ${\mathfrak {X}}$ is the data consisting of:

for each object $\xi$ , a quasi-coherent sheaf $F_{\xi }$ on the scheme $p(\xi )$ ,
for each morphism $H:\xi \to \eta$ in ${\mathfrak {X}}$ and $h=p(H):p(\xi )\to p(\eta )$ in the base category, an isomorphism
$\rho _{H}:h^{*}(F_{\eta }){\overset {\simeq }{\to }}F_{\xi }$

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}}{\overset {h_{1}^{*}(\rho _{H_{2}})}{\to }}h_{1}^{*}F_{\xi _{2}}{\overset {\rho _{H_{1}}}{\to }}F_{\xi _{1}}

equals

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.

Notes

Arbarello, Cornalba & Griffiths 2011, Ch. XIII., § 2.

References

Arbarello, Enrico; Griffiths, Phillip (2011). Geometry of algebraic curves. Vol. II, with a contribution by Joseph Daniel Harris. Grundlehren der mathematischen Wissenschaften. Vol. 268. doi:10.1007/978-3-540-69392-5. ISBN 978-3-540-42688-2. MR 2807457.
Behrend, Kai A. (2003). "Derived 𝑙-adic categories for algebraic stacks". Memoirs of the American Mathematical Society. 163 (774). doi:10.1090/memo/0774.
Laumon, Gérard; Moret-Bailly, Laurent (2000). Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Vol. 39. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-24899-6. ISBN 978-3-540-65761-3. MR 1771927.
Olsson, Martin (2007). "Sheaves on Artin stacks". Journal für die reine und angewandte Mathematik (Crelle's Journal). 2007 (603): 55–112. doi:10.1515/CRELLE.2007.012. S2CID 15445962. 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. doi:10.1093/imrn/rnv142.

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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[1] Arbarello, Cornalba & Griffiths 2011, Ch. XIII., § 2.