4
$\begingroup$

Hello,

suppose $R$ is a non-commutative ring of finite (left) global dimension and $M$ is a finitely generated (left) $R$-module.

So we know that there is a projective resolution of $M$ of finite length. The first term $P_0$ of the standard resolution will be finitely generated free. However, the next step would take into account the kernel $P_0\to M$, which need not be finitely generated.

So what about the general case? Is there always a resolution by finitely generated projective modules (allowing that specific resolution to be infinite)? If the answer is negative, which I would expect, what conditions on $R$ would make it true?

Thanks, D.

$\endgroup$
1
  • 6
    $\begingroup$ $R$ (left) Noetherian is both necessary and sufficient: if $R$ has a non-f.g. left ideal $I$ then the kernel of $R\to R/I$ will be $I$ and so not be finitely generated. The converse is not difficult. $\endgroup$ Commented Nov 16, 2012 at 10:33

2 Answers 2

6
$\begingroup$

Let me start recalling the Schanuel's Lemma:

If $M$ is a module and $P,P'$ are projective modules, then for every short exact sequences $0\to K\to P\to M\to 0$ and $0\to K'\to P'\to M\to 0$, there is an isomorphism $K\oplus P'\cong K'\oplus P$.

So, if you have a short exact sequence $0\to K\to P\to M\to 0$ with $M$ f.g., $P$ f.g. projective and $K$ not f.g., you will not be able to find any other f.g. projective $P'$ which admits a projection onto $M$ with f.g. kernel.

So, your question is equivalent to ask the following: for which class of rings does the class of f.g. left modules coincide with that of finitely presented left modules? The answer to this question is: the class of left Noetherian rings.

In some cases, one can find such resolutions outside from the Noetherian context. For example, if a ring is left coherent you can find such resolutions for any finitely generated left ideal of the ring.

$\endgroup$
1
  • 1
    $\begingroup$ And there was I looking at Schanuel's Lemma just minutes before asking, but didn't realise that it contains most of the answer. Thank you very much everyone! $\endgroup$
    – DaniW
    Commented Nov 16, 2012 at 12:13
5
$\begingroup$

The last sentence in the answer by Simone Virili can easily be generalised as follows:

If $R$ is left coherent, then a left $R$-module has a projective resolution whose components are of finite type if and only if it is of finite presentation.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.