Given a group $G$, a Ring $R$ and two $R[G]$-modules $M,N$. Then one can consider $Hom_R(M,N)$ and define inductively submodules $A_0,A_1,...$ via

$A_0:=0$

$A_{n+1}:=\{ \;f\; |\; \forall\; g\in G: (x\mapsto f(gx)-gf(x))\in A_n\}$

$A_1$ is then precisely $Hom_{R[G]}(M,N)$. So I am wondering, what the modules $A_2,..$ might be good for. A guess is, that this should be related to group cohomology in some way (, which I don't see).

For example with $G=\mathbb{Z}$ with generator $t$ and $M:=\mathbb{Z}[G],N:=\mathbb{Z}$ (with trivial $G$ action) one gets: $A_n=\{f \in Hom_\mathbb{Z}(M,N) | $ There are $ a_0,...,a_n \in \mathbb{Z} : f(t^n) =\sum_{i=0}^n a_i n^i\}$.

Given a group $G$, a ring $R$ and two $R[G]$-modules $M,N$. Then one can consider $Hom_R(M,N)$ and define inductively submodules $A_0,A_1,...$ via

$A_0:=0$

$A_{n+1}:=\{ \;f\; |\; \forall\; g\in G: (x\mapsto f(gx)-gf(x))\in A_n\}$

$A_1$ is then precisely $Hom_{R[G]}(M,N)$. So I am wondering, what the modules $A_2,..$ might be good for. A guess is, that this should be related to group cohomology in some way (, which I don't see).

For example with $G=\mathbb{Z}$ with generator $t$ and $M:=\mathbb{Z}[G],N:=\mathbb{Z}$ (with trivial $G$ action) one gets: $A_n=\{f \in Hom_\mathbb{Z}(M,N) | $ There are $ a_0,...,a_n \in \mathbb{Z} : f(t^n) =\sum_{i=0}^n a_i n^i\}$.