The Stacks project

Lemma 87.23.1. Let $A$ and $B$ be pre-adic topological rings. Let $\varphi : A \to B$ be a continuous ring homomorphism.

  1. If $\varphi $ is adic, then $\varphi $ is taut.

  2. If $B$ is complete, $A$ has a finitely generated ideal of definition, and $\varphi $ is taut, then $\varphi $ is adic.

In particular the conditions “$\varphi $ is adic” and “$\varphi $ is taut” are equivalent on the category $\textit{WAdm}^{adic*}$.

Proof. Part (1) is Lemma 87.6.4. Part (2) is Lemma 87.6.5. The final statement is a consequence of (1) and (2). $\square$

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