Timeline for Yoneda Ext theorem and extensions
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 14, 2023 at 17:50 | comment | added | Leo Alonso | @L.Xie Hope that with this edit, things a re a little bit more explicit. | |
| Apr 14, 2023 at 17:49 | history | edited | Leo Alonso | CC BY-SA 4.0 |
added 276 characters in body
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| Apr 10, 2023 at 22:54 | comment | added | L. Xie | @LeoAlonso An element in $Ext^n(M,N)$ corresponds to an $n$-extension that starts with $N$ and ends with $M$, right? Why is your extension starting with $M$ and ending with $N$? | |
| Apr 10, 2023 at 10:38 | comment | added | Leo Alonso | @L.Xie Indeed, it was a typo. The derived category perspective makes things easier, though. | |
| Apr 10, 2023 at 10:37 | history | edited | Leo Alonso | CC BY-SA 4.0 |
edited body
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| Apr 7, 2023 at 23:05 | comment | added | L. Xie | There are actually $n+1$ elements in between $M$ and $N$, this is an $ n+1$ extension, isn't it? | |
| Feb 3, 2022 at 18:51 | vote | accept | CommunityBot | moved from User.Id=30211 by developer User.Id=481663 | |
| Feb 3, 2022 at 18:50 | comment | added | user30211 | @LeoAlonso Thanks. I think I got it from your comment after all. | |
| Feb 3, 2022 at 18:40 | comment | added | Leo Alonso | @KindBubble Basically, there is a collection of isomorphism commuting with the identities of $N$ and $M$ (and among themselves). For more details see, for instance, Maclana, Homology, chapter III, §8, p. 96. | |
| Feb 3, 2022 at 18:18 | comment | added | user30211 | @LeoAlonso what is the equivalence relation on extensions? Is it weak equivalence? | |
| Feb 3, 2022 at 17:57 | comment | added | Leo Alonso | @KindBubble It is because injective resolutions are universal up to homotopy. | |
| Feb 3, 2022 at 17:57 | comment | added | Leo Alonso | @ZhenLin Of course, I was answering the case in the question. As a matter of fact, my argument work whenever the base category is Grothendieck. In general it is not clear to me that Ext and Yoneda-Ext agree | |
| Feb 3, 2022 at 17:46 | comment | added | user30211 | Sorry for not getting it yet, but how do the injective extensions correspond to ordinary ones? | |
| Feb 3, 2022 at 14:13 | comment | added | Zhen Lin | Out of curiosity: I think the theorem for $\textrm{Ext}^1$ can be proved without assuming enough injectives. The OP asks about modules over a ring but we could equally ask about any abelian category. | |
| Feb 3, 2022 at 12:38 | history | answered | Leo Alonso | CC BY-SA 4.0 |