Timeline for How to compute Ext-groups for categories without enough injectives/projectives?
Current License: CC BY-SA 4.0
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| Apr 15, 2020 at 9:34 | history | edited | YCor | CC BY-SA 4.0 |
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| Apr 15, 2020 at 9:20 | history | edited | Batominovski | CC BY-SA 4.0 |
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| Feb 26, 2018 at 8:47 | comment | added | Tim Campion | Also, if there is some formula relating the duality functor $(-)^\vee$ to $Hom$, you might be able to do something like a flat resolution... | |
| Feb 26, 2018 at 8:38 | comment | added | Tim Campion | You could try unpacking the usual proofs of the existence of derived functors and compute using projective / injective effacements. | |
| Mar 31, 2017 at 0:33 | comment | added | Ingo Blechschmidt | This comment probably won't help you, but it might help others with a similar question: A way of calculating Ext groups in Serre quotient categories is demonstrated by Mohamed Barakat and Markus Lange-Hegermann. | |
| Mar 30, 2017 at 21:51 | history | edited | Batominovski | CC BY-SA 3.0 |
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| Mar 30, 2017 at 21:45 | history | edited | Batominovski | CC BY-SA 3.0 |
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| Mar 30, 2017 at 21:39 | history | edited | Batominovski | CC BY-SA 3.0 |
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| Mar 30, 2017 at 19:37 | comment | added | math no more | Possibly useful: mathoverflow.net/questions/150847/semi-free-resolutions and references therein | |
| Mar 30, 2017 at 19:32 | comment | added | math no more | It $A$ is an abelian category, you can turn it into a dg category by taking the dg category of complexes in $A$. If $A$ is a dg category, then $D(A)$ is the homotopy category of a certain dg category (the idempotent completion of the essential image of the Yoneda embedding of $A$ into $Fun(A^{op}, Ch)$, dg-localized with respect to quasi-isomorphisms). In this localized category of modules I think you can compute Exts using semi-free resolutions, but I don't know a reference. Not sure if this makes sense or is overkill? | |
| Mar 30, 2017 at 17:29 | comment | added | Yemon Choi | To the OP: ah, you are using what I have sometimes seen denoted by Yext. (I thought this might be the case but wanted to check.) While I don't know the answer, perhaps some of the people on the category theory mailing list might have suggestions? This kind of foundational issue seems like it should have been considered in the 1960s and 1970s | |
| Mar 30, 2017 at 17:24 | comment | added | Yemon Choi | @DenisNardin Good point. I was thinking of cases where there is extra structure in one variable, i.e. Ext( _ , M) acylic where M has some good properties. E.g. calculation using flat resolutions | |
| Mar 30, 2017 at 16:50 | comment | added | Denis Nardin | @YemonChoi Unfortunately Ext-acyclics are projectives/injectives (depending of which variable you are deriving) so acyclic resolutions are not very useful in this context. | |
| Mar 30, 2017 at 16:44 | history | edited | Batominovski | CC BY-SA 3.0 |
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| Mar 30, 2017 at 16:21 | comment | added | Batominovski | @YemonChoi Ext-groups can be defined more generally in terms of Yoneda equivalence (of exact sequences), and it coincides with the definition we regularly use when there are enough injectives/projectives. See stacks.math.columbia.edu/tag/06XP, for example. I have a hard time using this definition to make any meaningful computation. | |
| Mar 30, 2017 at 15:14 | comment | added | Yemon Choi | How are you defining Ext, if not in terms of projective/injective resolutions? If I remember correctly, once one has defined Ext one can usually compute it in terms of acyclic resolutions, but this presupposes that it has a definition to begin with... | |
| Mar 30, 2017 at 15:03 | history | edited | Batominovski | CC BY-SA 3.0 |
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| Mar 30, 2017 at 13:31 | history | asked | Batominovski | CC BY-SA 3.0 |