Sheaf On An Algebraic Stack

(Algebraic Geometry)

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Algebraic Geometry
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What is Sheaf on an algebraic stack?

In algebraic geometry, a quasi-coherent sheaf on an algebraic stack is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is the data consists of, for each a scheme S in the base category and in , a quasi-coherent sheaf on S together with maps implementing the compatibility conditions among 's. For a Deligne–Mumford stack, there is a simpler description in terms of a presentation : a quasi-coherent sheaf on is one obtained by descending a quasi-coherent sheaf on U. A quasi-coherent sheaf on a Deligne–Mumford stack generalizes an (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.

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algebraic geometrysheaf theory

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Quasi-coherent sheaf on an algebraic stack

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