Given $G$ a group and $M$ a $G$-module, we denote by $(C(G,M),d)$ the cochain complex resulting from the standard resolution. An element in $H^n(G,M)$ can be written as the class $[a]$ of an element $a\in C^n(G,M)$.
I'm interested in how to adapt two results regarding the cohomology to cochains.
The first result is about the action of $G$ on $H(G,M)$, which is known to be trivial, i.e. $s[a]=[a]$ for all $[a]\in H^n(G,M)$ and $s\in G$. One proves that for all $a\in C^n(G,M)$ and $s\in G$ we have $$sa-a=(h_sd+dh_s)(a),$$ where $h_s:C(G,M)\to C(G,M)[-1]$ is an explicit linear map.
The second result is about the commutativity of the cup product, i.e. $[a]\cup [b]=(-1)^{pq}t_\ast ([b]\cup [a])$ for all $[a]\in H^p(G,M)$ and $[b]\in H^q(G,N)$. Here $t:M\otimes N\to N\otimes M$ denotes the natural bijection. One proves that for all $a\in H^p(G,M)$ and $b\in H^q(G,N)$ we have $$(-1)^{pq}t_*(b\cup a)-a\cup b=(hd+dh)(a\otimes b),$$ where $h:C(G,M)\otimes C(G,N)\to C(G,M\otimes N)[-1]$ is an explicit linear map.
I need these results as a prerequisite for a future paper. My question is whether any of these results is already known.
The first result appears in a clumsier way in an older question of mine: A formula that proves that $G$ acts trivially on $H^*(G,M)$
As you can see, if $a\in C^n(G,M)$, then $h_sa(s_1,\ldots,s_{n-1})=\sum_{k=0}^{n-1}(-1)^ka(s_1,\ldots,s_k,s,s^{-1}s_{k+1}s,\ldots,s^{-1}s_{n-1}s)$. In homogeneous cochains it looks a little better, $h_sa(s_1,\ldots,s_{n-1})=\sum_{k=0}^{n-1}(-1)^ka(s_0,\ldots,s_k,s_ks,\ldots,s_{n-1}s)$.
