In algebraic geometry, a quasi-coherent sheaf on an algebraic stack X {\displaystyle {\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 ξ {\displaystyle \xi } in X ( S ) {\displaystyle {\mathfrak {X}}(S)} , a quasi-coherent sheaf F ξ {\displaystyle F_{\xi }} on S together with maps implementing the compatibility conditions among F ξ {\displaystyle F_{\xi }} 's. For a Deligne–Mumford stack, there is a simpler description in terms of a presentation U → X {\displaystyle U\to {\mathfrak {X}}} : a quasi-coherent sheaf on X {\displaystyle {\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 (Arbarello, Cornalba & Griffiths 2011, Ch. XIII., Definition 2.1.) Let X {\displaystyle {\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 X {\displaystyle {\mathfrak {X}}} is the data consisting of:
for each object ξ {\displaystyle \xi } , a quasi-coherent sheaf F ξ {\displaystyle F_{\xi }} on the scheme p ( ξ ) {\displaystyle p(\xi )} , for each morphism H : ξ → η {\displaystyle H:\xi \to \eta } in X {\displaystyle {\mathfrak {X}}} and h = p ( H ) : p ( ξ ) → p ( η ) {\displaystyle h=p(H):p(\xi )\to p(\eta )} in the base category, an isomorphism
ρ H : h ∗ ( F η ) → ≃ F ξ {\displaystyle \rho _{H}:h^{*}(F_{\eta }){\overset {\simeq }{\to }}F_{\xi }}
satisfying the cocycle condition: for each pair H 1 : ξ 1 → ξ 2 , H 2 : ξ 2 → ξ 3 {\displaystyle H_{1}:\xi _{1}\to \xi _{2},H_{2}:\xi _{2}\to \xi _{3}} ,
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