Lemma 10.51.4 (Krull's intersection theorem). Let $R$ be a Noetherian local ring. Let $I \subset R$ be a proper ideal. Let $M$ be a finite $R$-module. Then $\bigcap _{n \geq 0} I^ nM = 0$.

**Proof.**
Let $N = \bigcap _{n \geq 0} I^ nM$. Then $N = I^ nM \cap N$ for all $n \geq 0$. By the Artin-Rees Lemma 10.51.2 we see that $N = I^ nM \cap N \subset IN$ for some suitably large $n$. By Nakayama's Lemma 10.20.1 we see that $N = 0$.
$\square$

## 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Your email address will not be published.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 #3856 by Alice on

Comment #3940 by Johan on