Generalization of traces

Hello all,

I already asked this question here, I hope it is ok to repeat it.

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.

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?

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.

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.

Thanks, D.

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Totally unrelated to your question, but you've got one too many http's in your linked url. It should read: math.stackexchange.com/questions/233602/generalized-traces –  Yuichiro Fujiwara Nov 10 '12 at 20:04
The category of $R$-modules for a noncommutative ring $R$ isn't automatically equipped with a monoidal structure, so let's pretend that you meant "commutative." Then, as you say, a trace can be defined for endomorphisms of dualizable objects in a closed symmetric monoidal category. $R$-modules are such a category. So are chain complexes of $R$-modules... On the other hand, it seems to me that there is no reasonable notion of trace of an arbitrary endomorphism on an infinite-dimensional vector space, so you should probably be more specific about your intended application. –  Qiaochu Yuan Nov 10 '12 at 21:00
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. –  DaniW Nov 10 '12 at 21:48
I think you want something like dualizable objects in the derived category for that. I don't know the details though. –  Qiaochu Yuan Nov 10 '12 at 22:57
No idea whether it is related to your question, but here is a reference: H. Rohrl, A categorical setting for determinants and traces, Nagoya Math. J. 34 (1969), 35-76 projecteuclid.org/euclid.nmj/1118797662 –  Pasha Zusmanovich Nov 14 '12 at 14:50