2008
Ault, Shaun
Fiedorowicz, Z.
The theory of symmetric homology, in which the symmetric groups op
k , for k 0, play the role that the cyclic groups do in cyclic homology, begins with the definition of the category S, containing the simplicial category as subcategory. Symmetric homology of a unital algebra, A, over commutative ground ring, k, is defined using derived functors and the symmetric bar construction of Fiedorowicz. If A = k[G] is a group ring, then HS(k[G]) is related to stable homotopy theory. Two chain complexes that compute HS(A)
are constructed, both making use of a symmetric monoidal category S+ containing S, which also permits homology operations to be defined on HS(A). Two spectral sequences are found that aid in computing symmetric homology. In the second spectral sequence, the complex Sym(p)
is constructed. This complex turns out to be isomorphic to the suspension
of the cycle-free chessboard complex, p+1, of Vre´cica and ˇZivaljevi´c. Recent results on the connectivity of n imply finite-dimensionality of the symmetric homology groups of finitedimensional algebras. Finally, an explicit partial resolution is presented, permitting the calculation of HS0(A) and HS1(A) for any finite-dimensional algebra A.1
k , for k 0, play the role that the cyclic groups do in cyclic homology, begins with the definition of the category S, containing the simplicial category as subcategory. Symmetric homology of a unital algebra, A, over commutative ground ring, k, is defined using derived functors and the symmetric bar construction of Fiedorowicz. If A = k[G] is a group ring, then HS(k[G]) is related to stable homotopy theory. Two chain complexes that compute HS(A)
are constructed, both making use of a symmetric monoidal category S+ containing S, which also permits homology operations to be defined on HS(A). Two spectral sequences are found that aid in computing symmetric homology. In the second spectral sequence, the complex Sym(p)
is constructed. This complex turns out to be isomorphic to the suspension
of the cycle-free chessboard complex, p+1, of Vre´cica and ˇZivaljevi´c. Recent results on the connectivity of n imply finite-dimensionality of the symmetric homology groups of finitedimensional algebras. Finally, an explicit partial resolution is presented, permitting the calculation of HS0(A) and HS1(A) for any finite-dimensional algebra A.1
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