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Closed Subscheme

A closed subscheme of a scheme X is a subscheme Z whose inclusion Z->X is a closed immersion. It is determined by an ideal sheaf I that is a quasi-coherent sheaf on X, with structure sheaf O_Z=O_X/I.

On an affine scheme X=Spec(R), the closed subscheme defined by an ideal I is Spec(R/I).

See also

Affine Scheme, Closed Immersion, Ideal Sheaf, Open Subscheme, Quasi-Coherent Sheaf, Structure 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. "Immersions of Schemes." §26.10 in The Stacks Project, Tag 01IM, 2026. https://stacks.math.columbia.edu/tag/01IM.

Cite this as:

Weisstein, Eric W. "Closed Subscheme." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ClosedSubscheme.html

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