Finitely generated resolutions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:57:37Z http://mathoverflow.net/feeds/question/112564 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112564/finitely-generated-resolutions Finitely generated resolutions DaniW 2012-11-16T10:16:19Z 2012-11-16T20:35:41Z <p>Hello,</p> <p>suppose $R$ is a non-commutative ring of finite (left) global dimension and $M$ is a finitely generated (left) $R$-module. </p> <p>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.</p> <p>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?</p> <p>Thanks, D.</p> http://mathoverflow.net/questions/112564/finitely-generated-resolutions/112565#112565 Answer by Simone Virili for Finitely generated resolutions Simone Virili 2012-11-16T10:33:27Z 2012-11-16T20:35:41Z <p>Let me start recalling the Schanuel's Lemma:</p> <p><em>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 $K\oplus P'\cong K'\oplus P$.</em></p> <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.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/112564/finitely-generated-resolutions/112568#112568 Answer by Fred Rohrer for Finitely generated resolutions Fred Rohrer 2012-11-16T11:07:06Z 2012-11-16T11:07:06Z <p>The last sentence in the answer by Simone Virili can easily be generalised as follows:</p> <p>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.</p>