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64.28 Profinite groups, cohomology and homology

Let $G$ be a profinite group.

Cohomology. Consider the category of discrete modules with continuous $G$-action. This category has enough injectives and we can define

\[ H^ i(G, M) = R^ iH^0(G, M) = R^ i(M\mapsto M^ G) \]

Also there is a derived version $RH^0(G, -)$.

Homology. Consider the category of compact abelian groups with continuous $G$-action. This category has enough projectives and we can define

\[ H_ i(G, M) = L_ iH_0(G, M)=L_ i(M\mapsto M_ G) \]

and there is also a derived version.

Trivial duality. The functor $M\mapsto M^\wedge = \mathop{\mathrm{Hom}}\nolimits _{cont}(M, S^1)$ exchanges the categories above and

\[ H^ i(G, M)^\wedge = H_ i(G, M^\wedge ) \]

Moreover, this functor maps torsion discrete $G$-modules to profinite continuous $G$-modules and vice versa, and if $M$ is either a discrete or profinite continuous $G$-module, then $M^\wedge = \mathop{\mathrm{Hom}}\nolimits (M, \mathbf{Q}/\mathbf{Z})$.

Notes on Homology.

  1. If we look at $\Lambda $-modules for a finite ring $\Lambda $ then we can identify

    \[ H_ i(G, M)=Tor_ i^{\Lambda [[G]]}(M, \Lambda ) \]

    where $\Lambda [[G]]$ is the limit of the group algebras of the finite quotients of $G$.

  2. If $G$ is a normal subgroup of $\Gamma $, and $\Gamma $ is also profinite then

    • $H^0(G, -)$: discrete $\Gamma $-module$\to $ discrete $\Gamma /G$-modules

    • $H_0(G, -)$: compact $\Gamma $-modules $\to $ compact $\Gamma /G$-modules

    and hence the profinite group $\Gamma /G$ acts on the cohomology groups of $G$ with values in a $\Gamma $-module. In other words, there are derived functors

    \[ RH^0(G, -) : D^{+}(\text{discrete }\Gamma \text{-modules}) \longrightarrow D^{+}(\text{discrete }\Gamma /G\text{-modules}) \]

    and similarly for $LH_0(G, -)$.

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