# 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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