Lemma 87.5.2 (0AMY)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.5: Taut ring maps
Lemma 87.5.2 (cite)

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Lemma 87.5.2. Let $\varphi : A \to B$ be a continuous map of weakly admissible topological rings. The following are equivalent

$\varphi$ is taut,
for every weak ideal of definition $I \subset A$ the closure of $\varphi (I)B$ is a weak ideal of definition of $B$ and these form a fundamental system of weak ideals of definition of $B$ .

Proof. The remarks following Definition 87.5.1 show that (2) implies (1). Conversely, assume $\varphi$ is taut. If $I \subset A$ is a weak ideal of definition, then the closure of $\varphi (I)B$ is open by definition of tautness and consists of topologically nilpotent elements by Lemma 87.4.10. Hence the closure of $\varphi (I)B$ is a weak ideal of definition. Furthermore, by definition of tautness these ideals form a fundamental system of open ideals and we see that (2) is true. $\square$

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