The Stacks project

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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