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 a data that 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

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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 on   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

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ℓ-adic formalism

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The ℓ-adic formalism (theory of ℓ-adic sheaves) extends to algebraic stacks.

See also

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  • 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

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References

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  • 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.
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