Timeline for Relation between $Tor_{C^*(BG;K)}(K,K) $ and $K^G$?
Current License: CC BY-SA 4.0
7 events
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| Jun 11, 2021 at 18:29 | comment | added | Maxime Ramzi | Ok then let me answer in the comments that for finite $p$-groups in characteristic $p$, the map is an equivalence, but not for $G=\mathbb Z_p$, more generally it won't be one for $G=$ the derived $p$-completion of some strictly smaller group . I'm not sure about the dual construction | |
| Jun 11, 2021 at 18:06 | comment | added | Hadrian Heine | For $K$ the algebraic closure of the field with $p$-elements ($p$ a prime) by a theorem of Mandell the map $\alpha$ as a map of $E_\infty$-algebras over $K$ goes to an equivalence under the functor sending an $E_\infty$-algebra $E$ over $K$ to the space of maps of $E_\infty$-algebras $E \to K.$ But I do not know how to understand the map $\alpha$ and $\beta$ from that. | |
| Jun 11, 2021 at 18:03 | history | edited | Hadrian Heine | CC BY-SA 4.0 |
added 24 characters in body
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| Jun 11, 2021 at 17:55 | comment | added | Hadrian Heine | @Maxime Ramzi: Actually, I am interested in the "dual" construction: the map $K[G] \to Cotor_{C_*(BG;K)}(K,K) $ and large abelian derived $p$-complete groups. But I try to first understand the "dual" situation of my question for finite $p$-groups and the $p$-adic numbers. | |
| Jun 11, 2021 at 17:53 | comment | added | Hadrian Heine | @Maxime Ramzi: I mean two things by "what can we say". 1. Under which conditions on the group is it an isomorphism? 2. Is there an interpretation of the map, especially in the situation, where the map is not an isomorphism? | |
| Jun 9, 2021 at 18:02 | comment | added | Maxime Ramzi | When you say "what can we say", do you mean more than "is it an equivalence ?". Also, are you more interested in small groups, e.g. finite or finitely generated, or in "bigger" groups ? I think I can answer for finite $p$-groups (not necessarily abelian) and for finitely generated ones | |
| Jun 9, 2021 at 17:25 | history | asked | Hadrian Heine | CC BY-SA 4.0 |