Lemma 87.4.11 (0AMZ)—The Stacks project

Processing math: 100%

Table of contents
Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.4: Topological rings and modules
Lemma 87.4.11 (cite)

numberstags

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. $\square$

Comments (0)

There are also:

2 comment(s) on Section 87.4: Topological rings and modules

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment