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2010 Symmetric homology of algebras
Shaun V Ault
Algebr. Geom. Topol. 10(4): 2343-2408 (2010). DOI: 10.2140/agt.2010.10.2343


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.


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Shaun V Ault. "Symmetric homology of algebras." Algebr. Geom. Topol. 10 (4) 2343 - 2408, 2010.


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

zbMATH: 07274347
MathSciNet: MR2748335
Digital Object Identifier: 10.2140/agt.2010.10.2343

Primary: 55N35
Secondary: 13D03 , 18G10

Keywords: bar construction , chessboard complex , Cyclic homology , ‎gap‎ , spectral sequence , symmetric homology

Rights: Copyright © 2010 Mathematical Sciences Publishers


Vol.10 • No. 4 • 2010
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