User daniw - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:32:01Z http://mathoverflow.net/feeds/user/27990 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/112017/generalization-of-traces Generalization of traces DaniW 2012-11-10T19:50:04Z 2012-11-10T20:08:15Z <p>Hello all,</p> <p>I already asked this question <a href="http://math.stackexchange.com/questions/233602/generalized-traces" rel="nofollow">here</a>, I hope it is ok to repeat it.</p> <p>A trace can be defined for endomorphisms of dualizable objects in a closed symmetric monoidal category. More concretely, in the category of $R$-modules for any associative ring $R$, a trace is defined for endomorphisms of finitely generated projective $R$-modules.</p> <p>The question can be stated in the more general setting from the beginning, but for simplicity: Is there a useful notion of a trace for (all!) endomorphisms of a more general class of modules than f.g.p. modules?</p> <p>I was thinking about something like: If $M$ is an $R$-module and $0\to P_n\to\dots\to P_1\to M\to0$ is a projective resolution of $M$ where every $P_k$ is not only projective but finitely generated, then $tr(f)$ can be defined by the formula $\sum(-1)^ktr(f_k)=0$, where the $f_k$ are an extension of $f$ and $f_0=f$. If I am not mistaken, this definition is independent of the resolution.</p> <p>Are there references for such a trace, or are their reasons that this does not make real sense? In particular at the moment I am unable to show/disprove that this trace is additive on short exact sequences, which certainly it should be.</p> <p>Thanks, D.</p> http://mathoverflow.net/questions/112564/finitely-generated-resolutions/112565#112565 Comment by DaniW DaniW 2012-11-16T12:13:23Z 2012-11-16T12:13:23Z 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! http://mathoverflow.net/questions/112017/generalization-of-traces Comment by DaniW DaniW 2012-11-14T18:51:52Z 2012-11-14T18:51:52Z @Pasha: Thanks a lot, that paper looks useful. I try to figure out what happens if I specialize to my intended case. http://mathoverflow.net/questions/112017/generalization-of-traces Comment by DaniW DaniW 2012-11-10T21:48:19Z 2012-11-10T21:48:19Z Of course you are right about the monoidal structure, nevertheless there is a trace for endomorphisms of $R$-modules with $R$ non-commutative (living in the abelianization of $R$). I was intending to get a trace for reasonably behaved finitely generated modules, so no infinite dimensions in sight. As I have sketched, modules with a finitely generated projective resolution seem promising candidates. As for applications, I would like to be able to pass to the homology of a fgp chain complex and obtain a trace in homology. This is usually not projective, so I need some weaker condition.