Closed immersion

Remarks

Example: for a ring $A$ and an ideal $I$, the morphism $\operatorname{Spec} A/I \to \operatorname{Spec} A$ induced by $A \to A/I$ is a closed immersion. Its image is the closed set $V(I)$, and on the stalk at $\mathfrak{p} \supseteq I$ the map is $A_{\mathfrak p} \to (A/I)_{\mathfrak p}$, which is onto. Every closed immersion into $\operatorname{Spec} A$ is isomorphic to one of these. Non-example: let $k$ be a field and let $f \colon \operatorname{Spec} k[t] \to \operatorname{Spec} k[t^2, t^3]$ come from the inclusion $k[t^2, t^3] \subseteq k[t]$. Then $f$ is a homeomorphism onto the whole space, so clause 1 holds. Clause 2 fails at the origin: $t$ does not lie in the image of the stalk map. Non-example: the open immersion $\operatorname{Spec} k[x, x^{-1}] \to \operatorname{Spec} k[x]$ fails clause 1, since its image $D(x)$ is not closed.