Closed immersion criterion for affine schemes ($\mathrm{Spec}B\hookrightarrow \mathrm{Spec}A⟺A\twoheadrightarrow B$)

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 ${\mathcal{O}}_{\mathrm{Spec}B}$ under an affine morphism is the sheaf associated with the $A$-module $B$: its sections on $D(a)$ are $B_{\varphi (a)}$. 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.