# An alternative description of normalized cochains in terms of tensor powers of the augmented ideal

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 $G\cong{\mathbb F}_p^r$

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.