Sheaf on an algebraic stack

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack 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 in , a quasi-coherent sheaf on S together with maps implementing the compatibility conditions among 's.

For a Deligne–Mumford stack, there is a simpler description in terms of a presentation : a quasi-coherent sheaf on 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 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 , a quasi-coherent sheaf on the scheme ,
  2. for each morphism in and in the base category, an isomorphism
satisfying the cocycle condition: for each pair ,
equals .

(cf. equivariant sheaf.)

Examples

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.

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.


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.