Lemma 87.23.1 (0GBS)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.23: Adic morphisms
Lemma 87.23.1 (cite)

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

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