nLab closed subscheme

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

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

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

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