Peiffer elements in simplicial groups and algebras

The main objective of this paper is to prove in full generality the following two facts:A. For an operad  OOin  AbAb, let  AAbe a simplicial  OO-algebra such that  AmAmis generated as an  OO-ideal by  (∑i=0m−1si(Am−1)), for  m>1m>1, and let  NAbe the Moore complex of  AA. Thend(NmA)=∑Iγ(Op⊗⋂i∈I1kerdi⊗⋯⊗⋂i∈Ipkerdi)where the sum runs over those partitions of  [m−1][m−1], I=(I1,…,Ip)I=(I1,…,Ip), p≥1p≥1, and  γγis the action of  OOon  AA.B. Let  GGbe a simplicial group with Moore complex  NGin which  GnGnis generated as a normal subgroup by the degenerate elements in dimension  n>1n>1, then  d(NnG)=∏I,J[⋂i∈Ikerdi,⋂j∈Jkerdj], for  I,J⊆[n−1]I,J⊆[n−1]with  I∪J=[n−1]I∪J=[n−1].In both cases, didi is the ii-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:AbΔop→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 NG of a simplicial group GG. This construction could be of interest in itself.