Lemma 87.7.4 (0GXH)—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 87.7.4 (cite)

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Lemma 87.7.4. Let $A$ be a weakly adic topological ring. Let $I$ be an ideal of definition such that $I/I_2$ is a finitely generated module where $I_2$ is the closure of $I^2$ . Then $A$ is adic and has a finitely generated ideal of definition.

Proof. We use the characterization of Lemma 87.7.2 without further mention. Choose $f_1, \ldots , f_ r \in I$ which map to generators of $I/I_2$ . Set $I' = (f_1, \ldots , f_ r)$ . We have $I' + I_2 = I$ . Then $I_2$ is the closure of $I^2 = (I' + I_2)^2 \subset I' + I_3$ where $I_3$ is the closure of $I^3$ . Hence $I' + I_3 = I$ . Continuing in this fashion we see that $I' + I_ n = I$ for all $n \geq 2$ where $I_ n$ is the closure of $I^ n$ . In other words, the closure of $I'$ in $A$ is $I$ . Hence the closure of $(I')^ n$ is $I_ n$ . Thus the closures of $(I')^ n$ are a fundamental system of open ideals of $A$ . We conclude by Lemma 87.7.3. $\square$

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