Symmetric homology of algebras

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 $A=k\left[\Gamma \right]$, the symmetric homology is related to stable homotopy theory via $H{S}_{\ast }\left(k\left[\Gamma \right]\right)\cong H_{\ast }\left(\Omega {\Omega }^{\infty }{S}^{\infty }\left(B\Gamma \right);k\right)$. Two chain complexes that compute $H{S}_{\ast }\left(A\right)$ are constructed, both making use of a symmetric monoidal category $\Delta S_{+}$ containing $\Delta S$. Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, ${Sym}_{\ast }^{\left(p\right)}$. ${Sym}^{\left(p\right)}$ is isomorphic to the suspension of the cycle-free chessboard complex ${\Omega }_{p+1}$ of Vrećica and Živaljević, and so recent results on the connectivity of ${\Omega }_n$ imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the $k{\Sigma }_{p+1}$–module structure of ${Sym}^{\left(p\right)}$ are devloped. A partial resolution is found that allows computation of $H{S}_1\left(A\right)$ for finite-dimensional $A$ and some concrete computations are included.