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Let $K$ be an algebraically closed field and $G$ a group.

Given a dg-algebra $A$, a left $A$-module $M$ and a right $A$-module $N$ let $Tor_A(M,N)$ denote the homology of the derived tensor product $M \otimes^L_A N,$ i.e. the realization of the Bar-construction $B(M,A,N) $ with $B(M,A,N)_n = M \otimes_K A^{\otimes n} \otimes N$.

There is a canonical map $\alpha: K \otimes^L_{C^*(BG;K)} K \to C^*(G;K) $ of dg-algebras over $K$ (in fact of $E_\infty$-algebras over $K$) that induces on homology a graded map $$ \beta: Tor_{C^*(BG;K)}(K,K) \to K^G. $$

What can we say about the map $\beta$ if $G$ is a derived $p$-complete abelian group, say for example $\mathbb{Z}^\wedge_p$ or $\mathbb{Z}/p^n \mathbb{Z}?$

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  • $\begingroup$ 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 $\endgroup$
    – Maxime Ramzi
    Commented Jun 9, 2021 at 18:02
  • $\begingroup$ @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? $\endgroup$
    – Hadrian Heine
    Commented Jun 11, 2021 at 17:53
  • $\begingroup$ @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. $\endgroup$
    – Hadrian Heine
    Commented Jun 11, 2021 at 17:55
  • $\begingroup$ 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. $\endgroup$
    – Hadrian Heine
    Commented Jun 11, 2021 at 18:06
  • $\begingroup$ 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 $\endgroup$
    – Maxime Ramzi
    Commented Jun 11, 2021 at 18:29

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