Peiffer elements in simplicial groups and algebras

J.L. CASTIGLIONI; M. LADRA

JOURNAL OF PURE AND APPLIED ALGEBRA

Elsevier

Año: 2008 p. 2115 - 2128

0022-4049

The main objective of this paper is to proof in all generalitythe following two facts:\vspace{2pt} oindent A. emph{ For an operad $oo$ in $ab$, let$A$ be a simplicial $oo$-algebra such that $A_m$ is generated asan $oo$-ideal by $(sum_{i = 0}^{m-1} s_i(A_{m-1}))$, for $m >1$, and let $N A$ be the Moore complex of $A$. Then[d (N_m A) = sum_{I } gamma(oo_{p} otimes igcap_{i inI_1}ker d_i otimes dots otimes igcap_{i in I_{p}}ker d_i)] where the sum runs over those partitions of $[m-1]$,$I = (I_1,dots,I_p)$, $p geq 1$ , and $gamma$ is the action of$oo$ on $A$.}\vspace{3pt} oindent B. emph{ Let $G$ be a simplicial group withMoore complex $N G$ in which $G_n$ is generated as a normalsubgroup of $G_n$ by the degenerate elements in dimension $n >1$,then $d(N_nG) = prod_{I,J}[igcap_{i in I}ker d_i, igcap_{jin J}kerd_j]$, for $I,J subseteq [n-1]$ with $I cup J = [n-1]$.}\vspace{2pt} oindent In both cases, $d_i$ is the $i-th$ face ofthe corresponding simplicial object. The former result completes and generalizes results from Akc{c}aand Arvasi cite{AA}, and Arvasi and Porter cite{AP}; the lattercompletes a result from Mutlu and Porter cite{MP}. Our approach tothe problem is different from that of the cited works. We have firstsucceeded with a proof for the case of algebras over an operad byintroducing a different description of the inverse of thenormalization functor $N: sab o ch$. For the case of simplicialgroups, we have then adapted the construction for the inverseequivalence used for algebras to get a simplicial group $N Goxtimes lb$ from the Moore complex $N G$ of a simplicial group$G$. This construction could be of interest in itself.