Symmetric homology of algebras

Ault, Shaun V

doi:10.2140/agt.2010.10.2343

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Symmetric homology of algebras

Shaun V Ault

Algebraic & Geometric Topology 10 (2010) 2343–2408

DOI: 10.2140/agt.2010.10.2343

Abstract

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 $H S_{*} (k [Γ]) ≅ H_{*} (Ω Ω^{\infty} S^{\infty} (B Γ); k)$ . Two chain complexes that compute $H S_{*} (A)$ are constructed, both making use of a symmetric monoidal category $Δ S_{+}$ containing $Δ S$ . 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 $Ω_{p + 1}$ of Vrećica and Živaljević, and so recent results on the connectivity of $Ω_{n}$ imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the $k Σ_{p + 1}$ –module structure of ${Sym}^{(p)}$ are devloped. A partial resolution is found that allows computation of $H S_{1} (A)$ for finite-dimensional $A$ and some concrete computations are included.

Keywords

symmetric homology, bar construction, spectral sequence, chessboard complex, GAP, cyclic homology

Mathematical Subject Classification 2000

Primary: 55N35

Secondary: 13D03, 18G10

References

Publication

Received: 6 January 2010

Accepted: 17 July 2010

Published: 19 December 2010

Authors

Shaun V Ault
Department of Mathematics Fordham University Bronx NY 10461 USA
http://fordham.academia.edu/ShaunAult/

Volume 10, issue 4 (2010)