Finitely generated resolutions

Question

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.

Answer 1

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.

Answer 2

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.