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 $\N A$ be the Moore complex of $A$. Then \[ d (\N_m A) = \sum_{I} \gamma(\oo_{p} \otimes \bigcap_{i \in I_1}\ker d_i \otimes ... \otimes \bigcap_{i \in I_{p}}\ker d_i) \] where the sum runs over those partitions of $[m-1]$, $I = (I_1,...,I_p)$, $p \geq 1$, and $\gamma$ is the action of $\oo$ on $A$. B. Let $G$ be a simplicial group with Moore complex $\N G$ in which the normal subgroup of $G_n$ generated by the degenerate elements in dimension $n$ is the proper $G_n$. Then $d(\N_nG) = \prod_{I,J}[\bigcap_{i \in I}\ker d_i, \bigcap_{j \in J}\ker d_j]$, 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 Ak\c{c}a 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.