nLab
closed subscheme

A morphism $i=(i,i^{♯}):Y\hookrightarrow X$ 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 $i^{♯}:{𝒪}_X\to f_{*}{𝒪}_Y$ 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 $i\prime :Y\prime \to X$ are equivalent if there is an isomorphism $\mathrm{eq}:Y\cong Y\prime$ of schemes such that $i\prime \circ \mathrm{eq}=i$)

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

In the affine case, one can take $X=\mathrm{Spec}R$ and $Y=\mathrm{Spec}R/I$ where $I={\Gamma }_X(ℐ)$ 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 $I$.

See also open subscheme.