subscheme

A subscheme of a scheme X is a scheme Z equipped with a morphism i: Z → X that is a closed immersion, identifying Z with a closed subset of X and carrying a compatible O_X-algebra structure. Equivalently, a subscheme is specified by a sheaf of ideals I ⊆ O_X that is quasi-coherent; the closed subscheme Z defined by I has underlying topological space V(I) and structure sheaf O_Z ≅ i^{-1} O_X / I i^{-1} O_X. In this sense, a subscheme records not only a subset of X but also a scheme structure on that subset.

In the affine case, if X = Spec A, every closed subscheme corresponds to a quotient A → A/I, and Z ≅ Spec(A/I). The inclusion i: Z → X is induced by this quotient, and the closed subset is V(I) in Spec A.

Subschemes come in several related flavors. A closed subscheme arises from a sheaf of ideals on X. An open subscheme is obtained by restricting to an open subset U ⊆ X, with the inclusion U → X an open immersion. A locally closed subscheme is a scheme structure on a locally closed subset that is compatible with a closed immersion into X. Every subscheme can be covered by affine open subschemes, reflecting that subschemes can be studied locally.

Additional terminology: a subscheme is reduced if its defining ideal I is radical; nonreduced subschemes carry nilpotent structure sheaves reflecting embedded or multiplicity information. Subschemes are fundamental for defining intersections, supports of sheaves, and various geometric properties within a global scheme.