Volume 10, issue 4 (2010)

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Symmetric homology of algebras

Shaun V Ault

Algebraic & Geometric Topology 10 (2010) 2343–2408

The symmetric homology of a unital algebra A over a commutative ground ring k is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring A = k[Γ], the symmetric homology is related to stable homotopy theory via HS(k[Γ])H(ΩΩS(BΓ);k). Two chain complexes that compute HS(A) are constructed, both making use of a symmetric monoidal category ΔS+ containing ΔS. Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, Sym(p). Sym(p) is isomorphic to the suspension of the cycle-free chessboard complex Ωp+1 of Vrećica and Živaljević, and so recent results on the connectivity of Ωn imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the kΣp+1–module structure of Sym(p) are devloped. A partial resolution is found that allows computation of HS1(A) for finite-dimensional A and some concrete computations are included.

symmetric homology, bar construction, spectral sequence, chessboard complex, GAP, cyclic homology
Mathematical Subject Classification 2000
Primary: 55N35
Secondary: 13D03, 18G10
Received: 6 January 2010
Accepted: 17 July 2010
Published: 19 December 2010
Shaun V Ault
Department of Mathematics
Fordham University
Bronx NY 10461