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From page 101 in Atiyah-MacDonald:

"Two of the important properties of localization are that it preserves exactness and the Noetherian property...."

I remember proving that it preserves exactness, it's proposition 3.3. on page 39. But what is meant by "Noetherian property"? Thanks.

Rudy the Reindeer
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1 Answers1

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It means that if $M$ is Noetherian then so is $S^{-1}M$. If you first prove that for any submodule $N'$ of $S^{-1}M$ one has $N' = S^{-1}N$, where $N$ is the inverse image of $N'$ in $M$, then this follows quickly.

Dylan Moreland
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  • Thank you, I didn't realise it meant that. For rings this is theorem 7.3. on page 80. – Rudy the Reindeer Jul 25 '12 at 06:28
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    @Clark Yep, it's common to prove that (and a few other facts about extension/contraction along $M \to S^{-1}M$) for rings, but it's also true for modules. For this fact I don't see any difference in the proof, really. – Dylan Moreland Jul 25 '12 at 06:30
  • Sure, I just wanted to add the comment in case someone else reads this thread and is interested. : ) – Rudy the Reindeer Jul 25 '12 at 06:31