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.