The Stacks project

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

  1. $R \to S$ 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.

  2. $R \to S$ 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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