closed subscheme

A morphism i=(i,i ):YXi = (i,i^\sharp):Y\hookrightarrow X of algebraic schemes is a closed immersion if, as a topological space, i(Y)i(Y) is a closed subspace of XX, ii is a homeomorphism on the image, and the comorphism i :𝒪 Xi *𝒪 Yi^\sharp:\mathcal{O}_X\to i_*\mathcal{O}_Y is a surjective map of rings.

A closed subscheme of XX is an equivalence class of closed immersions into XX (morphisms of schemes i:YXi:Y\to X and i:YXi':Y'\to X are equivalent if there is an isomorphism eq:YYeq:Y\cong Y' of schemes such that ieq=ii'\circ eq = i)

In an equivalent description of closed subschemes, there is a sheaf of ideals 𝒪 X\mathcal{I}\subset\mathcal{O}_X such that the quotient sheaf 𝒪 X/\mathcal{O}_X/\mathcal{I} is supported on the image set i(Y)Yi(Y)\cong Y. One says that \mathcal{I} is the defining sheaf of ideals (or in more French style, the ideal of definition) of the subscheme YY. The structure sheaf 𝒪 Y\mathcal{O}_Y of the subscheme YY is the quotient sheaf 𝒪 X/\mathcal{O}_X/\mathcal{I} restricted to YY. The conormal sheaf of the closed immersion YXY\hookrightarrow X is the quasicoherent sheaf of modules defined by formula / 2\mathcal{I}/\mathcal{I}^2.

In the affine case, one can take X=SpecRX=Spec\,R and Y=SpecR/IY=Spec\,R/I where I=Γ X()I=\Gamma_X(\mathcal{I}) is the defining ideal of the closed affine subscheme YY. The underlying set of YY is precisely the set V(I)V(I) of all prime ideals in RR which contain II.

See also open subscheme.

Last revised on November 28, 2013 at 06:15:39. See the history of this page for a list of all contributions to it.