Quasi-coherent Sheaf On An Algebraic Stack

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack $\mathfrak$ 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 $\mathfrak(S)$, a quasi-coherent sheaf $F_$ on ''S'' together with maps implementing the compatibility conditions among $F_$'s.

For a Deligne–Mumford stack, there is a simpler description in terms of a presentation $U \to \mathfrak$: a quasi-coherent sheaf on $\mathfrak$ 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 $\mathfrak$ 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 $\mathfrak$ is the data consisting of:
# for each object $\xi$, a quasi-coherent sheaf $F_$ on the scheme $p(\xi)$,
# for each morphism $H: \xi \to \eta$ in $\mathfrak$ and $h = p(H): p(\xi) \to p(\eta)$ in the base category, an isomorphism
#:$\rho_H: h^*(F_) \overset\to F_$
:satisfying the cocycle condition: for each pair $H_1: \xi_1 \to \xi_2, H_2: \xi_2 \to \xi_3$,
::$h_1^* h_2^* F_ \overset \to h_1^* F_ \overset\to F_$ equals $h_1^* h_2^* F_ \overset= (h_2 \circ h_1)^* F_ \overset\to F_$.
(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)

Notes

References

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* Editorial note: This paper corrects a mistake in Laumon and Moret-Bailly's ''Champs algébriques''.
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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

Sheaf theory
Algebraic geometry

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