## Algebraic & Geometric Topology

### Symmetric homology of algebras

Shaun V Ault

#### 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 $HS∗(k[Γ])≅H∗(ΩΩ∞S∞(BΓ);k)$. 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
Accepted: 17 July 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715173

Digital Object Identifier
doi:10.2140/agt.2010.10.2343

Mathematical Reviews number (MathSciNet)
MR2748335

#### 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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