Closed subscheme

A subscheme of a scheme $ X $ defined by a quasi-coherent sheaf of ideals $ J $ of the structure sheaf $ {\mathcal O} _ {X} $ as follows: The topological space of the subscheme, $ V ( J ) $, is the support of the quotient sheaf $ {\mathcal O} _ {X} / J $, and the structure sheaf is the restriction of $ {\mathcal O} _ {X} / J $ to its support. A morphism of schemes $ f : Y \rightarrow X $ is called a closed imbedding if $ f $ is an isomorphism of $ Y $ onto some closed subscheme in $ X $; a closed imbedding is a monomorphism in the category of schemes. For any closed subset $ Y \subset X $ there exists a minimal closed subscheme in $ X $ with space $ Y $, known as the reduced closed subscheme with space $ Y $. If $ Y $ is a subscheme of $ X $, then the smallest closed subscheme $ Y _ {1} $ of $ X $ containing $ Y $ is known as the (schematic) closure of the subscheme $ Y $ in $ X $.

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001