= Closed immersion criterion for affine schemes
{title2=$\operatorname{Spec}B\hookrightarrow\operatorname{Spec}A\iff A\twoheadrightarrow B$}
A surjective <ring homomorphism> $A\to B=A/I$ identifies the <spectrum of a ring> $B$ with the closed subset $V(I)$ and induces a surjective map of structure sheaves. Conversely, the <direct image sheaf> of $\mathcal O_{\operatorname{Spec}B}$ under an affine morphism is the sheaf associated with the $A$-module $B$: its sections on $D(a)$ are $B_{\phi(a)}$. The cokernel of the structure-sheaf map is therefore associated with the module cokernel of $A\to B$. If the sheaf map is surjective, all localizations of that cokernel vanish and the module cokernel is zero. This proves the converse without assuming global sections preserve arbitrary sheaf surjections.
Back to article page
