Sheaf on an algebraic stack

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.

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