I want to know if the following alternative of the normalized non-homogeneous cochains is already know.

Let $G$ be a group and let ${\mathcal I}={\mathcal I}_G$ be its augmentation ideal, ${\mathcal I}=\ker (\varepsilon :{\mathbb Z}[G]\to{\mathbb Z})$.

If $M$ is a $G$-module then let $C^n(G,M)=\{ a:G^n\to M\, :\, a(u_1,\ldots,u_n)=0\text{ if }u_i=1\text{ for some }i\}$ be the normalized non-homogeneous cochains of degree $n$.

Definition We define the ${\mathcal I}$-cochains of degree $n$ as $$C_{\mathcal I}^n(G,M):={\rm Hom}(T^n({\mathcal I}),M).$$

We have an isomorphism $C_{\mathcal I}^n(G,M)\to C^n(G,M)$ given by $f\mapsto a$, where $$a(u_1,\ldots,u_n)=f((u_1-1)\otimes\cdots\otimes (u_n-1)).$$

Then the coboundary map $d_n:C^n(G,M)\to C^{n+1}(G,M)$, translated in the language of ${\mathcal I}$-cochains writes in a very convenient form. Namely, we have:

Proposition The coboundary map $d_n:C_{\mathcal I}^n(G,M)\to C_{\mathcal I}^{n+1}(G,M)$ is given by \begin{multline*} d_nf(\alpha_1\otimes\cdots\otimes\alpha_{n+1})\\ =\alpha_1f(\alpha_2\otimes\cdots\otimes\alpha_{n+1})+\sum_{i=1}^n(-1)^if(\alpha_1\otimes\cdots\otimes\alpha_i\alpha_{i+1}\otimes\cdots\otimes\alpha_{n+1}) \end{multline*} for every $\alpha_1,\ldots,\alpha_{n+1}\in{\mathcal I}$.

This formula looks very similar to the one for $d_n:C^n(G,M)\to C^{n+1}(G,M)$, but with the last term of $d_na(u_1,\ldots,u_{n+1})$, $(-1)^{n+1}a(u_1,\ldots,u_n)$, ignored.

The closest thing I found is in Hilton-Stammbach, chapter VI, 13(c). ("Alternative Description of the Bar Resolution".) If we denote by $\bar C^n(G,M)$ the normalized homogeneous cochains and by $\bar C_{\mathcal I}^n(G,M)={\rm Hom}_G({\mathbb Z}\otimes T^n({\mathcal I}),M)$ the cochains resulting from the alternative description of the bar reductions, then the element $\bar a\in\bar C^n(G,M)$ corresponds to $\bar f\in\bar C_{\mathcal I}^n(G,M)$ if $\bar a(u_0,\ldots,u_{n+1})=\bar f(u_0\otimes (u_1-u_0)\otimes\cdots\otimes (u_n-u_{n-1}))$ $\forall u_0,\ldots,u_n\in G$.

The element $f\in C_{\mathcal I}^n(G,M)$ corresponding to $\bar f\in\bar C_{\mathcal I}^n(G,M)$ is given by $f((u_1-1)\otimes\cdots\otimes (u_n-1))=\bar f(1\otimes (u_1-1)\otimes u_1(u_2-1)\otimes\cdots\otimes u_1\cdots u_{n-1}(u_n-1))$ $\forall u_1,\ldots,u_n\in G$. As one can see, there is no nice, simple relation between $f\in C_{\mathcal I}^n(G,M)$ and $\bar f\in C_{\mathcal I}^n(G,M)$, such as, say, $f(\eta )=\bar f(1\otimes\eta )$ $\forall\eta\in T^n({\mathcal I})$.

I mention that I used these ${\mathcal I}$-cochains to solve the problem I described here:

Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$


1 Answer 1


I have just posted an article on arXiv: https://arxiv.org/abs/2005.11868

The first section deals with these ${\mathcal I}$-cochains. Again, if you saw them somewhere, maybe with some other name and notation, then please let me know.


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