Article ID Journal Published Year Pages File Type
4598089 Journal of Pure and Applied Algebra 2008 14 Pages PDF
Abstract

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.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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