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 $A = k[\Gamma]$, the symmetric homology is related to stable homotopy theory via $HS_*(k[\Gamma]) \cong H_*(\Omega\Omega^{\infty} S^{\infty}(B\Gamma); k)$. Two chain complexes that compute $HS_*(A)$ 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, $Sym^{(p)}_*$. $Sym^{(p)}$ is isomorphic to the suspension of the cycle-free chessboard complex $\Omega_{p+1}$ of Vre\'{c}ica and \v{Z}ivaljevi\'{c}, 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 $Sym^{(p)}$ are devloped. A partial resolution is found that allows computation of $HS_1(A)$ for finite-dimensional $A$ and some concrete computations are included.