Group cohomology via monads or triple cohomology

Group cohomology via monads or triple cohomology

[Johannes Huebschmann]

Dear All

Let $R$ be a commutative ring,
$G$ a group,
$\mathrm{Mod}_R$ the category of $R$-modules,
$\mathrm{Mod}_{RG}$ that of right $RG$-modules,
let
\[
\mathcal G\colon \mathrm{Mod}_{R}\longrightarrow \mathrm{Mod}_{RG}
\]
be the familiar functor which assigns to the $R$-module $V$
the right $RG$-module $\mathrm{Map}(G, V)$, with right $G$-structure
being given by left translation in $G$, and let
\[
\square \colon \mathrm{Mod}_{RG}\longrightarrow \mathrm{Mod}_{R}
\]
be the forgetful functor.
The unit of the resulting adjunction is well known to be given by
the assignment to the right $RG$-module $V$ of
\[
\eta_V\colon V \longrightarrow \mathrm{Map}(G, \square V),
\ v \longmapsto \eta_v:G \to  \square V,\ \eta_v(x) =vx, \ v \in V, x \in
G.
\]
Given the $RG$-module $V$,
the standard construction associated with $V$ and the resulting monad
$(T,\eta,\mu)$ yields an injective resolution of $V$ in the category
of  right
$RG$-modules. All this is entirely standard and classical.

Consider instead the functor
\[
\mathrm{Mod}_{RG}\longrightarrow \mathrm{Mod}_{RG}
\]
which assigns to the right $RG$-module $V$
the right $RG$-module $\mathrm{Map}(G, V)$,
with right $G$-structure being given by

diagonal action

relative to left translation in $G$
and replace $\eta$ with $\omega$ given by
the assignment to the right $RG$-module $V$ of
\[
\omega_V\colon V \longrightarrow \mathrm{Map}(G, V),
\ v \longmapsto \omega_v:G \to V,\ \omega_v(x) =v, \ v \in V, x
\in G.
\]
These data, together with the appropriate natural transformation
replacing the composition $\mu$,
yield an alternate description of the monad $(T,\eta,\mu)$.
This has certainly been discussed in detail in the literature.
I am looking for a precise reference.

Many thanks in advance

Johannes


HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
http://math.univ-lille1.fr/~huebschm

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