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Locally Closed Subscheme

A locally closed subscheme of a scheme X is a closed subscheme of an open subscheme of X. Equivalently, its inclusion into X is a scheme immersion.

The underlying set of a locally closed subscheme is therefore the intersection of an open set and a closed set of X.

See also

Closed Immersion, Open Immersion, Scheme Immersion, 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. "Locally Closed Subscheme." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LocallyClosedSubscheme.html

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