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reference request – An different description of normalized cochains when it comes to tensor powers of the augmented mannequin retort

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reference request – An different description of normalized cochains when it comes to tensor powers of the augmented mannequin

I need to know if the next different of the normalized non-homogeneous cochains is already know.

Let $G$ breathe a bunch and let ${mathcal I}={mathcal I}_G$ breathe its augmentation mannequin, ${mathcal I}=ker (varepsilon :{mathbb Z}[G]to{mathbb Z})$.

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

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

We maintain an isomorphism $C_{mathcal I}^n(G,M)to C^n(G,M)$ given by $fmapsto a$, the place
$$a(u_1,ldots,u_n)=f((u_1-1)otimescdotsotimes (u_n-1)).$$

Then the coboundary map $d_n:C^n(G,M)to C^{n+1}(G,M)$, translated within the language of ${mathcal I}$-cochains writes in a really handy design. Namely, we maintain:

Proposition The coboundary map $d_n:C_{mathcal I}^n(G,M)to C_{mathcal I}^{n+1}(G,M)$ is given by
commence{multline*}
d_nf(alpha_1otimescdotsotimesalpha_{n+1})
=alpha_1f(alpha_2otimescdotsotimesalpha_{n+1})+sum_{i=1}^n(-1)^if(alpha_1otimescdotsotimesalpha_ialpha_{i+1}otimescdotsotimesalpha_{n+1})
stop{multline*}

for each $alpha_1,ldots,alpha_{n+1}in{mathcal I}$.

This system seems similar to the one for $d_n:C^n(G,M)to C^{n+1}(G,M)$, however with the ultimate time period of $d_na(u_1,ldots,u_{n+1})$, $(-1)^{n+1}a(u_1,ldots,u_n)$, ignored.

The closest factor I create is in Hilton-Stammbach, chapter VI, 13(c). (“Alternative Description of the barrier Resolution“.) If we denote by $barrier C^n(G,M)$ the normalized homogeneous cochains and by $barrier C_{mathcal I}^n(G,M)={rm Hom}_G({mathbb Z}otimes T^n({mathcal I}),M)$ the cochains ensuing from the choice description of the barrier reductions, then the part $barrier ainbarrier C^n(G,M)$ corresponds to $barrier finbarrier C_{mathcal I}^n(G,M)$ if $barrier a(u_0,ldots,u_{n+1})=barrier f(u_0otimes (u_1-u_0)otimescdotsotimes (u_n-u_{n-1}))$ $forall u_0,ldots,u_nin G$.

The part $fin C_{mathcal I}^n(G,M)$ akin to $barrier finbarrier C_{mathcal I}^n(G,M)$ is given by $f((u_1-1)otimescdotsotimes (u_n-1))=barrier f(1otimes (u_1-1)otimes u_1(u_2-1)otimescdotsotimes u_1cdots u_{n-1}(u_n-1))$ $forall u_1,ldots,u_nin G$. As one can graze, there is no such thing as a good, unostentatious relation between $fin C_{mathcal I}^n(G,M)$ and $barrier fin C_{mathcal I}^n(G,M)$, resembling, say, $f(eta )=barrier f(1otimeseta )$ $foralletain T^n({mathcal I})$.

I point out that I used these ${mathcal I}$-cochains to decipher the issue I described right here:

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

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