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Closed subscheme

From Encyclopedia of Mathematics
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A subscheme of a scheme defined by a quasi-coherent sheaf of ideals of the structure sheaf as follows: The topological space of the subscheme, , is the support of the quotient sheaf , and the structure sheaf is the restriction of to its support. A morphism of schemes is called a closed imbedding if is an isomorphism of onto some closed subscheme in ; a closed imbedding is a monomorphism in the category of schemes. For any closed subset there exists a minimal closed subscheme in with space , known as the reduced closed subscheme with space . If is a subscheme of , then the smallest closed subscheme of containing is known as the (schematic) closure of the subscheme in .


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References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Closed subscheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed_subscheme&oldid=17023
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article