Peiffer elements in simplicial groups and algebras

• Autores: Castiglioni, José Luis; Ladra, M.
• Año de publicación: 2008
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Ciencias Exactas; Matemática; Peiffer elements; algebras; simplicial groups
• Institución: Universidad Nacional de La Plata

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_m is generated as an O-ideal by (∑_{i = 0}^m-1 s_i (A_m-1)), for m > 1, and let NA be the Moore complex of A. Then d(N_mA) = ∑_Iγ (Op⊗ ∩ _i∈I1 ker d_i ⊗ ⋯ ⊗ ∩ _i∈Ip ker d_i) where the sum runs over those partitions of [m - 1], I = (I₁, ..., I_p), 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_n is generated as a normal subgroup by the degenerate elements in dimensionn > 1, then d (N_nG) = ∏I, J [∩_i∈I ker d_i, ∩_i∈J ker d_j], for I, J ⊆ [n - 1] with I ∪ 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 [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^Δop → 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.
Facultad de Ciencias Exactas