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 $N = 0$. $\square$

## Comments (2)

Comment #3856 by Alice on

While minor, It would be better to simply say $N=IN$, insteady of the roundabout $N\subset IN$. Also a note, the only point where locality is used is to guarente $I \subset \mathop{rad} (R)$ for all I for nakayama's lemma, so this can be generalised very easily if so desired.

Comment #3940 by on

Of course the application of Artin-Rees 10.51.2 does not immediately give what was said in the proof of the lemma or what you say. What I mean is that you have to think about it! So we can leave it as is I think for now.

The point about non-local cases is made in Remark 10.51.6 for the module equal to the ring. But yeah, we could discuss the case of a module there too.

Further discussion: The interesting part of the Artin-Rees lemma, and the way most people probably think about it, is that $I^nM \cap N$ is contained in $IN$ for $n \gg 0$. The fact that in the Artin-Rees lemma, as stated in the Stacks project and many texts, you get an equality is a bit of a red herring and rarely useful. This is why you'll see most often the Artin-Rees lemma used by saying $I^nM \cap N \subset IN$ for some $n > 0$.

This whole section can probably be improved upon and extended. For example, do lemmas 10.51.2 and 10.51.3 really cover all possible reformulations of Artin-Rees? Etc.

## 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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00IP. Beware of the difference between the letter 'O' and the digit '0'.