The symmetric homology of a unital algebra over a commutative ground ring is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring , the symmetric homology is related to stable homotopy theory via . Two chain complexes that compute are constructed, both making use of a symmetric monoidal category containing . Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, . is isomorphic to the suspension of the cycle-free chessboard complex of Vrećica and Živaljević, and so recent results on the connectivity of imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the –module structure of are devloped. A partial resolution is found that allows computation of for finite-dimensional and some concrete computations are included.
Shaun V Ault. "Symmetric homology of algebras." Algebr. Geom. Topol. 10 (4) 2343 - 2408, 2010. https://doi.org/10.2140/agt.2010.10.2343