Lemma 10.41.3 (00HW)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.41: Going up and going down
Lemma 10.41.3 (cite)

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Lemma 10.41.3. Let $R \to S$ be a ring map.

$R \to S$ satisfies going down if and only if generalizations lift along the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ , see Topology, Definition 5.19.4.
$R \to S$ satisfies going up if and only if specializations lift along the map $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ , see Topology, Definition 5.19.4.

Proof. Omitted. $\square$

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