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The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor: $$FIX: \mathcal{M_G} \to \mathcal{Ab}$$ where $FIX$ is the functor from the category of $G$-modules to the category of Abelian groups sending each module $M$ to its subgroup consisting of all elements of $M$ which are fixed by $G$ action.

Now we fix a natural number $n\in \mathbb{N}$ and do an obvious generalization of the above construction:

Instead of the above left exact functor $FIX$ we consider the functor $P_n$ which send a $G$-midule $M$ to the following subgroup of $M$: $$P_n(M)=\{x\in M \mid g.x= nx, \forall g \in G\}$$.

In this way what would be the corresponding derived functor?What kind of cohomology theory would appear? Is this a trivial generalization of classical "Group Cohomology"?If not, what would be a topological analogy?

The later question is motivated by the fact that the group cohomology corresponds to singular cohomology of the corresponding Eilenberg-Mclane space)

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    $\begingroup$ This is just fixed points in the $(n-1)$-torsion of $M$, or else $\operatorname{Hom}_G(\mathbb Z/(n-1)\mathbb Z,M)$, with $\mathbb Z/(n-1)\mathbb Z$ having the trivial $G$-action; accordingly you will just get $\operatorname{Ext}$ for that module. It should be easy to relate this to a mixture of "ordinary" Ext and "ordinary" $G$-cohomology $\endgroup$ Apr 8, 2018 at 20:45
  • $\begingroup$ Is $g\in G$ fixed? $\endgroup$ Apr 9, 2018 at 2:10
  • $\begingroup$ @JulianRosen non $g\in G$ varies over $G$.I revise the question. $\endgroup$ Apr 9, 2018 at 15:46
  • $\begingroup$ @მამუკაჯიბლაძე thank you for your comment and your attention to the question. $\endgroup$ Apr 9, 2018 at 15:47

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