Lemma 90.4.7. Let $R$ be an object of $\widehat{\mathcal{C}}_\Lambda $. Let $(J_ n)$ be a decreasing sequence of ideals such that $\mathfrak m_ R^ n \subset J_ n$. Set $J = \bigcap J_ n$. Then the sequence $(J_ n/J)$ defines the $\mathfrak m_{R/J}$-adic topology on $R/J$.

**Proof.**
It is clear that $\mathfrak m_{R/J}^ n \subset J_ n/J$. Thus it suffices to show that for every $n$ there exists an $N$ such that $J_ N/J \subset \mathfrak m_{R/J}^ n$. This is equivalent to $J_ N \subset \mathfrak m_ R^ n + J$. For each $n$ the ring $R/\mathfrak m_ R^ n$ is Artinian, hence there exists a $N_ n$ such that

Set $E_ n = (J_{N_ n} + \mathfrak m_ R^ n)/\mathfrak m_ R^ n$. Set $E = \mathop{\mathrm{lim}}\nolimits E_ n \subset \mathop{\mathrm{lim}}\nolimits R/\mathfrak m_ R^ n = R$. Note that $E \subset J$ as for any $f \in E$ and any $m$ we have $f \in J_ m + \mathfrak m_ R^ n$ for all $n \gg 0$, so $f \in J_ m$ by Krull's intersection theorem, see Algebra, Lemma 10.51.4. Since the transition maps $E_ n \to E_{n - 1}$ are all surjective, we see that $J$ surjects onto $E_ n$. Hence for $N = N_ n$ works. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #7732 by Mingchen on

Comment #7982 by Stacks Project on