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.
