A closed subscheme of is an equivalence class of closed immersions into (morphisms of schemes and are equivalent if there is an isomorphism of schemes such that )
In an equivalent description of closed subschemes, there is a sheaf of ideals such that the quotient sheaf is supported on the image set . One says that is the defining sheaf of ideals (or in more French style, the ideal of definition) of the subscheme . The structure sheaf of the subscheme is the quotient sheaf restricted to . The conormal sheaf of the closed immersion is the quasicoherent sheaf of modules defined by formula .
In the affine case, one can take and where is the defining ideal of the closed affine subscheme . The underlying set of is precisely the set of all prime ideals in which contain .
See also open subscheme.