The Stacks project

Lemma 26.18.5. Let $f : X \to S$ be a morphism of schemes. Consider the diagrams

\[ \xymatrix{ X_ s \ar[r] \ar[d] & X \ar[d] & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \times _ S X \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(\kappa (s)) \ar[r] & S & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \ar[r] & S } \]

In both cases the top horizontal arrow is a homeomorphism onto its image.

Proof. Choose an open affine $U \subset S$ that contains $s$. The bottom horizontal morphisms factor through $U$, see Lemma 26.13.1 for example. Thus we may assume that $S$ is affine. If $X$ is also affine, then the result follows from Algebra, Remark 10.17.8. In the general case the result follows by covering $X$ by open affines. $\square$

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