Sheaf on an algebraic stack

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack $$\mathfrak{X}$$ 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 $$\mathfrak{X}(S)$$, 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 $$U \to \mathfrak{X}$$: a quasi-coherent sheaf on $$\mathfrak{X}$$ 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 $$\mathfrak{X}$$ 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 $$\mathfrak{X}$$ is the data consisting of:
 * 1) for each object $$\xi$$, a quasi-coherent sheaf $$F_{\xi}$$ on the scheme $$p(\xi)$$,
 * 2) for each morphism $$H: \xi \to \eta$$ in $$\mathfrak{X}$$ and $$h = p(H): p(\xi) \to p(\eta)$$ in the base category, an isomorphism
 * $$\rho_H: h^*(F_{\eta}) \overset{\simeq}\to F_{\xi}$$
 * satisfying the cocycle condition: for each pair $$H_1: \xi_1 \to \xi_2, H_2: \xi_2 \to \xi_3$$,
 * $$h_1^* h_2^* F_{\xi_3} \overset{h_1^* (\rho_{H_2})} \to h_1^* F_{\xi_2} \overset{\rho_{H_1}}\to F_{\xi_1}$$ equals $$h_1^* h_2^* F_{\xi_3} \overset{\sim}= (h_2 \circ h_1)^* F_{\xi_3} \overset{\rho_{H_2 \circ H_1}}\to F_{\xi_1}$$.

(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.