Sheaf on an algebraic stack

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In algebraic geometry, a quasi-coherent sheaf on an algebraic stack [math]\displaystyle{ \mathfrak{X} }[/math] 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 [math]\displaystyle{ \xi }[/math] in [math]\displaystyle{ \mathfrak{X}(S) }[/math], a quasi-coherent sheaf [math]\displaystyle{ F_{\xi} }[/math] on S together with maps implementing the compatibility conditions among [math]\displaystyle{ F_{\xi} }[/math]'s. For a Deligne–Mumford stack, there is a simpler description in terms of a presentation [math]\displaystyle{ U \to \mathfrak{X} }[/math]: a quasi-coherent sheaf on [math]\displaystyle{ \mathfrak{X} }[/math] 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)

Let [math]\displaystyle{ \mathfrak{X} }[/math] 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 [math]\displaystyle{ \mathfrak{X} }[/math] is the data consisting of:

  1. for each object [math]\displaystyle{ \xi }[/math], a quasi-coherent sheaf [math]\displaystyle{ F_{\xi} }[/math] on the scheme [math]\displaystyle{ p(\xi) }[/math],
  2. for each morphism [math]\displaystyle{ H: \xi \to \eta }[/math] in [math]\displaystyle{ \mathfrak{X} }[/math] and [math]\displaystyle{ h = p(H): p(\xi) \to p(\eta) }[/math] in the base category, an isomorphism
    [math]\displaystyle{ \rho_H: h^*(F_{\eta}) \overset{\simeq}\to F_{\xi} }[/math]
satisfying the cocycle condition: for each pair [math]\displaystyle{ H_1: \xi_1 \to \xi_2, H_2: \xi_2 \to \xi_3 }[/math],
[math]\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} }[/math] equals [math]\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} }[/math].

(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.

References

  • Arbarello, Enrico; Griffiths, Phillip (2011). Geometry of algebraic curves. Vol. II, with a contribution by Joseph Daniel Harris. Grundlehren der mathematischen Wissenschaften. 268. doi:10.1007/978-3-540-69392-5. ISBN 978-3-540-42688-2. 
  • 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. 39. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-24899-6. ISBN 978-3-540-65761-3. 
  • 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.  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. doi:10.1093/imrn/rnv142. 

External links




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