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The Stacks project

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

  1. $\varphi $ is taut,

  2. 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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