Time

Jul 29 2008 - 1:30pm - 3:30 pm

Location

MW 154

Speaker

Shaun Ault (OSU)

Seminar Website

http://www.math.ohio-state.edu/~ault/Thesis.pdf

Abstract

The theory of symmetric homology, in which the symmetric groups $\Sigma_k^{op}$ for $k\ge 0$, play the role that the cyclic groups do in cyclic homology, begins with the definition of the category $\Delta S$, containing the simplicial category $\Delta$ as subcategory. 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. 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 $\Delta S_+$ containing $\Delta S$, 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, $\Omega^{p+1}$, of Vrecica and Zivaljevic. Recent results on the connectivity of $\Omega_n$ imply finite-dimensionality of the symmetric homology groups of finite dimensional algebras. Finally, an explicit partial resolution is presented, permitting the calculation of $HS_0(A)$ and $HS_1(A)$ for a finite-dimensional algebra A.