Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C^*(M,G),d)$ the complex of inhomogeneous cochains, i.e. $C^n(G,M)=M^{G^n}$.

We say that a cochain $a\in C^n(G,M)$ is multilinear if it is linear in each variable, i.e. if for every $1\leq i\leq n$ and every $s_1,\ldots,s'_i,s''_i,\ldots,s_n\in G$ we have $$a(s_1,\ldots,s'_is''_i,\ldots,s_n)=a(s_1,\ldots,s'_i,\ldots,s_n)+a(s_1,\ldots,s''_i,\ldots,s_n).$$

I'm interested in the cochains with multilinear differentials, i.e. in the set $$\{ a\in C^n(G,M):da\text{ is multilinear}\}.$$

Of course, these cochains are more general than the usual cocycles, for which $da=0$.

My question is whether this kind of cochains appears somewhere in the literature. Maybe not in the general case, maybe just in some specific situations. Anything related to multiplicative cochains or to cochains with multilinear differentials would help.



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