Lemma 10.35.22 (00GE)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.35: Jacobson rings
Lemma 10.35.22 (cite)

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Lemma 10.35.22. With notation as above. Assume that $R$ is a Noetherian Jacobson ring. Further assume $R \to S$ is of finite type. There is a commutative diagram

\[ \xymatrix{ \text{Constr}(Y) \ar[r]^{E \mapsto E_0} \ar[d]^{E \mapsto f(E)} & \text{Constr}(Y_0) \ar[d]^{E \mapsto f(E)} \\ \text{Constr}(X) \ar[r]^{E \mapsto E_0} & \text{Constr}(X_0) } \]

where the horizontal arrows are the bijections from Topology, Lemma 5.18.8.

Proof. Since $R \to S$ is of finite type, it is of finite presentation, see Lemma 10.31.4. Thus the image of a constructible set in $X$ is constructible in $Y$ by Chevalley's theorem (Theorem 10.29.10). Combined with Lemma 10.35.21 the lemma follows. $\square$

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