Timeline for A Hom-Tensor identity - $\text{Hom}_{R}(P,B)\otimes _SC \cong \text{Hom}_{R}(P,B \otimes_S C) $
Current License: CC BY-SA 3.0
11 events
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| Mar 3, 2016 at 11:53 | comment | added | Jeremy Rickard | @DietrichBurde Though closely related, this is not the same question as the one on MSE. | |
| Mar 2, 2016 at 19:37 | comment | added | Dietrich Burde | Crossposted at MSE. | |
| Feb 29, 2016 at 22:11 | answer | added | Jeremy Rickard | timeline score: 6 | |
| Feb 29, 2016 at 13:59 | history | edited | tzelin1016 | CC BY-SA 3.0 |
Assume that $P$ is finitely-generated $R$-module.
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| Feb 29, 2016 at 13:43 | comment | added | tzelin1016 | @Jeremy Rickard Thank you very much for your question. Sorry for this unclear point. That is, $P$ is finitely generated over $R$. But I am also interested in any condition that will make this an natural isomorphism. (For example, $\mathcal{C}$ comes from some (not necessarily full) abelian subcategory of other module category (say, modules over algebra $T$) in which the corresponding $P$ is finite generated $T$-module. But I know this don't make sense yet...) Thanks again! | |
| Feb 29, 2016 at 13:30 | comment | added | Jeremy Rickard | @tzelin1016 Could you clarify what you mean by $P$ being "finitely generated"? Do you mean finitely generated as an $R$-module? Or finitely generated in some sense as an object of $\mathcal{C}$ (in which case could you say exactly what you mean by that)? | |
| Feb 29, 2016 at 13:13 | comment | added | tzelin1016 | Thank you very much for your comments. But I am still confused about the isomorphism $P^*\otimes_R - \cong \text{hom}_R(P,-)$ on $\mathcal{C}$ since $\mathcal{C}$ is an arbitrary full abelian subcategory. ($P$ is only projective in $\mathcal{C}$). | |
| Feb 29, 2016 at 13:02 | history | edited | tzelin1016 | CC BY-SA 3.0 |
added 19 characters in body
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| Feb 29, 2016 at 13:02 | comment | added | Todd Trimble | I'm not certain everyone will believe the question belongs here (as opposed to Mathematics.StackExchange), but the easiest way is probably to observe that for $P$ finitely generated projective, we have a canonical natural isomorphism $P^\ast \otimes_R - \cong \hom_R(P, -)$, from which the statement follows by associativity of tensor products of bimodules. You can drop the finiteness assumption on $C$. | |
| Feb 29, 2016 at 12:56 | review | First posts | |||
| Feb 29, 2016 at 13:23 | |||||
| Feb 29, 2016 at 12:56 | history | asked | tzelin1016 | CC BY-SA 3.0 |