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Let $G$ a free abelian group with $n$ generators $t_{1},t_{2},...,t_{n}$ and $M$ be a $G$-module. Is there an explicit description of the cohomology groups $H^{i}(G,M), i=0,1,2....$ ?

It is well know the case when $n=1$.

I will appreciate a reference in order to know that description.

Arturo Magidin
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José Luis Camarillo Nava
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    When $G$ acts trivially on $M$, the cohomology of $G$ agrees with the cohomology of the classifying space of $G$ (which, for $\mathbb{Z}^n$ is the $n$-torus $\mathbb{T}^n$). Then $H^k(\mathbb{Z}^n,M) \cong H^k(\mathbb{T}^n,M) \cong M^{\binom{n}{k}}$. In fact, if $R$ is a ring (which $G$ acts on trivially) then this isomorphism is true on the level of graded algebras as well, and $H^\bullet(\mathbb{Z}^n, R) \cong H^\bullet(\mathbb{T}^n,R) \cong \Lambda^n R$ (the nth exterior algebra of $R$). See, eg. section 3.2 in Hatcher for the relevant computation. – Chris Grossack Aug 27 '21 at 08:19
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    When $G$ doesn't act trivially on $M$, the answer seems to be more complicated... It's something to do with "local systems" on the $n$-torus (see here or here for instance). I'm sure that this is also an easy computation for someone who knows what local systems are, but alas I'm not that person (at least at time of writing)... Hopefully someone else can fill that gap. – Chris Grossack Aug 27 '21 at 08:23

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