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Quasi-Coherent Sheaf


A quasi-coherent sheaf on a scheme X is a sheaf F of O_X-modules such that, on every open subscheme U=Spec(R) that is an affine scheme, its restriction has the form M^~ for an R-module M.

Quasi-coherent sheaves translate module constructions into geometric data. In particular, quasi-coherent ideal sheaves correspond to closed subschemes.


See also

Affine Scheme, Ideal Sheaf, Module, Open Subscheme, Sheaf, Subscheme

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References

Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. I. Le langage des schémas." Publ. Math. IHES 4, 5-228, 1960. https://doi.org/10.1007/BF02684778.The Stacks Project Authors. "Quasi-Coherent Modules." §17.10 in The Stacks Project, Tag 01BD, 2026. https://stacks.math.columbia.edu/tag/01BD.

Cite this as:

Weisstein, Eric W. "Quasi-Coherent Sheaf." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Quasi-CoherentSheaf.html

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