Lemma 87.7.5 (0GXI)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.7: Weakly adic rings
Lemma 87.7.5 (cite)

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Lemma 87.7.5. Let $\varphi : A \to B$ be a continuous homomorphism of linearly topologized rings. If $\varphi$ is taut and $A$ is weakly pre-adic, then $B$ is weakly pre-adic.

Proof. Let $I \subset A$ be an ideal such that the closure $I_ n$ of $I^ n$ is open and these closures define a fundamental system of open ideals. Then the closure of $I^ nB$ is equal to the closure of $I_ nB$ . Since $\varphi$ is taut, these closures are open and form a fundamental system of open ideals of $B$ . Hence $B$ is weakly pre-adic. $\square$

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