The Stacks project

Lemma 87.4.11. Let $A \to B$ be a continuous map of linearly topologized rings. Let $I \subset A$ be an ideal. The closure of $IB$ is the kernel of $B \to B \widehat{\otimes }_ A A/I$.

Proof. Let $J_\mu $ be a fundamental system of open ideals of $B$. The closure of $IB$ is $\bigcap (IB + J_\lambda )$ by Lemma 87.4.2. Let $I_\mu $ be a fundamental system of open ideals in $A$. Then

\[ B \widehat{\otimes }_ A A/I = \mathop{\mathrm{lim}}\nolimits (B/J_\lambda \otimes _ A A/(I_\mu + I)) = \mathop{\mathrm{lim}}\nolimits B/(J_\lambda + I_\mu B + I B) \]

Since $A \to B$ is continuous, for every $\lambda $ there is a $\mu $ such that $I_\mu B \subset J_\lambda $, see discussion in Example 87.4.1. Hence the limit can be written as $\mathop{\mathrm{lim}}\nolimits B/(J_\lambda + IB)$ and the result is clear. $\square$

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