Peiffer elements in simplicial groups and algebras

Abstract

The main objective of this paper is to prove in full generality the following two facts: A. For an operad O in Ab, let A be a simplicial O-algebra such that A<SUB>m</SUB> is generated as an O-ideal by (∑<SUB>i = 0</SUB><SUP>m-1</SUP> s<SUB>i</SUB> (A<SUB>m-1</SUB>)), for m > 1, and let NA be the Moore complex of A. Then d(N<SUB>m</SUB>A) = ∑<SUB>I</SUB>γ (Op⊗ ∩ <SUB>i∈I₁</SUB> ker d<SUB>i</SUB> ⊗ ⋯ ⊗ ∩ <SUB>i∈I_p</SUB> ker d<SUB>i</SUB>) where the sum runs over those partitions of [m - 1], I = (I<SUB>1</SUB>, ..., I<SUB>p</SUB>), p ≥ 1, and γ is the action of O on A. B. Let G be a simplicial group with Moore complex NG in which G<SUB>n</SUB> is generated as a normal subgroup by the degenerate elements in dimensionn > 1, then d (N<SUB>n</SUB>G) = ∏I, J [∩<SUB>i∈I</SUB> ker d<SUB>i</SUB>, ∩<SUB>i∈J</SUB> ker d<SUB>j</SUB>], for I, J ⊆ [n - 1] with I ∪ J = [n - 1]. In both cases, d<SUB>i</SUB> is the i-th face of the corresponding simplicial object. The former result completes and generalizes results from Akça and Arvasi [I. Akça, Z. Arvasi, Simplicial and crossed Lie algebras, Homology Homotopy Appl. 4 (1) (2002) 43-57], and Arvasi and Porter [Z. Arvasi, T. Porter, Higher dimensional Peiffer elements in simplicial commutative algebras, Theory Appl. Categ. 3 (1) (1997) 1-23]; the latter completes a result from Mutlu and Porter [A. Mutlu, T. Porter, Applications of Peiffer pairings in the Moore complex of a simplicial group, Theory Appl. Categ. 4 (7) (1998) 148-173]. 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 inverse of the normalization functor N:Ab<SUP>Δ^op</SUP> → Ch≥ 0. For the case of simplicial groups, we have then adapted the construction for the inverse equivalence used for algebras to get a simplicial group NG ⊠ Λ from the Moore complex N G of a simplicial group G. This construction could be of interest in itself.