Algebraic & Geometric Topology

Symmetric homology of algebras

Shaun V Ault

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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 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.

Article information

Source
Algebr. Geom. Topol., Volume 10, Number 4 (2010), 2343-2408.

Dates
Received: 6 January 2010
Accepted: 17 July 2010
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715173

Digital Object Identifier
doi:10.2140/agt.2010.10.2343

Mathematical Reviews number (MathSciNet)
MR2748335

Subjects
Primary: 55N35: Other homology theories
Secondary: 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 18G10: Resolutions; derived functors [See also 13D02, 16E05, 18E25]

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

Citation

Ault, Shaun V. Symmetric homology of algebras. Algebr. Geom. Topol. 10 (2010), no. 4, 2343--2408. doi:10.2140/agt.2010.10.2343. https://projecteuclid.org/euclid.agt/1513715173


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