Timeline for Generalization of traces
Current License: CC BY-SA 3.0
9 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Nov 14, 2012 at 18:51 | comment | added | DaniW | @Pasha: Thanks a lot, that paper looks useful. I try to figure out what happens if I specialize to my intended case. | |
| Nov 14, 2012 at 14:50 | comment | added | Pasha Zusmanovich | 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 | |
| Nov 10, 2012 at 22:57 | comment | added | Qiaochu Yuan | I think you want something like dualizable objects in the derived category for that. I don't know the details though. | |
| Nov 10, 2012 at 21:48 | comment | added | DaniW | 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. | |
| Nov 10, 2012 at 21:00 | comment | added | Qiaochu Yuan | 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. | |
| Nov 10, 2012 at 20:08 | history | edited | DaniW | CC BY-SA 3.0 |
link corrected
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| Nov 10, 2012 at 20:04 | comment | added | Yuichiro Fujiwara | 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 | |
| Nov 10, 2012 at 19:50 | history | asked | DaniW | CC BY-SA 3.0 |