Peiffer Elements in Simplicial Groups and Algebras

Abstract

The main objectives of this paper are to give general proofs of the following two facts: A. For an operad $\oo$ in $\ab$, let $A$ be a simplicial $\oo$-algebra such that $A_m$ is the $\oo$-subalgebra generated by $({\sum }_{i=0}^m{s}_i(A_{m-1}))$, for every $n$, and let $\mathbb{N}A$ be the Moore complex of $A$. Then \[ d (\N_m A) = \sum_{I} γ(\oo_{p} \otimes \bigcap_{i \in I_1}\ker d_i \otimes ... \otimes \bigcap_{i \in I_{p}}\ker d_i) \] where the sum runs over those partitions of $[m-1]$, $I=(I_1,...,I_p)$, $p\ge 1$, and $\gamma$ is the action of $\oo$ on $A$. B. Let $G$ be a simplicial group with Moore complex $\mathbb{N}G$ in which the normal subgroup of $G_n$ generated by the degenerate elements in dimension $n$ is the proper $G_n$. Then $d({\mathbb{N}}_{n}G)={\prod }_{I,J}[{\bigcap }_{i\in I}\ker d_i,{\bigcap }_{j\in J}\ker d_j]$, for $I,J\subseteq [n-1]$ with $I\cup J=[n-1]$. In both cases, $d_i$ is the $i-th$ face of the corresponding simplicial object. The former result completes and generalizes results from Akça and Arvasi, and Arvasi and Porter; the latter, results from Mutlu and Porter. Our approach to the problem is different from that of the cited works. We have first succeeded with a proof for the case of algebras over an operad by introducing a different description of the adjoint inverse of the normalization functor $\N: \sab \to \ch$. For the case of simplicial groups, we have then adapted the construction for the adjoint inverse used for algebras to get a simplicial group $G \boxtimes \lb$ from the Moore complex of a simplicial group $G$. This construction could be of interest in itself.

18 pages