Timeline for On the difference between a projective chain complex and a level-wise projective chain complex
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 3, 2012 at 10:20 | comment | added | Zhen Lin | No, not at all. I'm remarking that for this functor $F$, the existence of a zero object is precisely what allows us to construct a simultaneous left and right adjoint. Try it and see! | |
| Aug 3, 2012 at 8:57 | comment | added | Shlomi A | If I understand you correctly, you claim that if $F$ is a functor between two categories which have a zero object, then a right adjoint of $F$ is also a left adjoint of it? Why is that so? Could you please formulate it precisely, and indicate how it could be proved? | |
| Aug 1, 2012 at 18:13 | comment | added | Zhen Lin | Because $0$ is both initial and terminal, we have both $F \dashv U$ and $U \dashv F$. A highly unusual situation, to be sure! | |
| Aug 1, 2012 at 11:34 | comment | added | Shlomi A | Hi Zhen. After considering it, I agree that for an adjoint pair, if the right adjoint preserves epimorphisms, then the left adjoint preserves projectives. However, in the example you have given, $U$ is the left adjoint, and $F$ the right one (and not as written), so you must have meant $U\dashv F$. If I'm not mistaken, it follows from this argument that a projective $R$-module $M$ gives a projective object $U(M)$ in $Ch(R)$, and not in the other way around. | |
| Aug 1, 2012 at 11:10 | vote | accept | Shlomi A | ||
| Aug 1, 2012 at 11:30 | |||||
| Jul 31, 2012 at 13:31 | vote | accept | Shlomi A | ||
| Jul 31, 2012 at 13:31 | |||||
| Jul 31, 2012 at 9:05 | history | answered | Zhen Lin | CC BY-SA 3.0 |