Timeline for Quasi-isomorphism preserves group hypercohomology
Current License: CC BY-SA 4.0
11 events
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| Sep 3, 2021 at 15:08 | comment | added | Dylan Wilson | It sounds like you're interested in crossed modules more generally. In that case, why not look at the classifying space (i.e. the equivalent data of the 2-type)? Then your condition of 'quasi-isomorphism' gives a weak equivalence of 2-types (indeed: the corresponding 2-type is the fiber of a map of 1-types, and your condition is exactly that you have a commuting square and that the induced map on the fiber is an equivalence on pi_1 and pi_2). Now any invariant of the 2-type is a 'quasi-isomorphism invariant'. For example, the cohomology of that classifying space. | |
| Sep 3, 2021 at 14:52 | answer | added | Mikhail Borovoi | timeline score: 0 | |
| Sep 3, 2021 at 13:43 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
I removed the last edit in order to put it into my answer
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| Sep 3, 2021 at 13:33 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
I have added my definitions of cohomology and hypercohomology for a a group of order2 and a complex of length 2.
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| Sep 1, 2021 at 16:58 | answer | added | Dan Petersen | timeline score: 5 | |
| Sep 1, 2021 at 16:38 | comment | added | Denis Nardin | What's your definition of group hypercohomology? The definition I usually use makes this true by definition... | |
| Sep 1, 2021 at 15:10 | comment | added | Mikhail Borovoi | @AndyPutman: This book also contains sign rules for the differentials in double complexes obtained as ${\rm Hom}(C,C')$ and $C\otimes C'$, where $C$ and $C'$ are given complexes. Many thanks again! | |
| Sep 1, 2021 at 15:06 | comment | added | Mikhail Borovoi | @AndyPutman: Many thanks! The necessary assertion is stated in Brown's book (for group homology rather than cohomology). This is Proposition VII.5.2: Weak equivalence of chain $G$-complexes induces isomorphism on homology. | |
| Sep 1, 2021 at 14:31 | history | edited | Mikhail Borovoi | CC BY-SA 4.0 |
edited body
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| Sep 1, 2021 at 14:31 | comment | added | Andy Putman | I don’t think this is stated there, but Brown’s book on group cohomology defines cohomology with coefficients in a chain complex and constructs a natural spectral sequence converging to it. Your quasi-isomorphism induces an isomorphism between these spectral sequences. There might be a more explicit reference, but that might do if you can’t find one. | |
| Sep 1, 2021 at 14:22 | history | asked | Mikhail Borovoi | CC BY-SA 4.0 |