Sheaf on an algebraic stack explained

ak{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 ak{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 ak{X} : a quasi-coherent sheaf on ak{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

Let

ak{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 ak{X} is the data consisting of:

1. for each object

\xi , a quasi-coherent sheaf F _{\xi} on the scheme p(\xi) ,

1. for each morphism

H: \xi \to η in ak{X} and h = p(H): p(\xi) \to p( η ) in the base category, an isomorphism\rho _{H:} h ^{*(F} _{η} ) \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_ 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_.(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

• Book: 10.1007/978-3-540-69392-5. Geometry of algebraic curves. Vol. II, with a contribution by Joseph Daniel Harris . Grundlehren der mathematischen Wissenschaften . 2011 . 268 . 978-3-540-42688-2. Arbarello . Enrico . Griffiths . Phillip. 2807457. .
• 10.1090/memo/0774. Derived -adic categories for algebraic stacks . 2003 . Behrend . Kai A. . Memoirs of the American Mathematical Society . 163 . 774 . free .
• Book: Laumon . Gérard . Gérard Laumon. Moret-Bailly . Laurent . Champs algébriques . . Berlin, New York . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics . 978-3-540-65761-3 . 1771927 . 2000 . 39. 10.1007/978-3-540-24899-6.
• 10.1515/CRELLE.2007.012. Sheaves on Artin stacks . 2007 . Olsson . Martin . Journal für die reine und angewandte Mathematik (Crelle's Journal) . 2007 . 603 . 15445962. 55–112 . Editorial note: This paper corrects a mistake in Laumon and Moret-Bailly's Champs algébriques.
• 10.1093/imrn/rnv142. Approximation of Sheaves on Algebraic Stacks . 2016 . Rydh . David . International Mathematics Research Notices . 2016 . 3 . 717–737. 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