Gamma (co)homology of commutative algebras and some related representations of the symmetric group

Abstract

This thesis covers two related subjects: homology of commutative algebras and certain representations of the symmetric group. There are several different formulations of commutative algebra homology, all of which are known to agree when one works over a field of characteristic zero. During 1991-1992 my supervisor, Dr. Alan Robinson, motivated by homotopy-theoretic ideas, developed a new theory, Γ-homology [Rob, 2]. This is a homology theory for commutative rings, and more generally rings commutative up to homotopy. We consider the algebraic version of the theory. Chapter I covers background material and Chapter II describes Γ-homology. We arrive at a spectral sequence for Γ-homology, involving objects called tree spaces. Chapter III is devoted to consideration of the case where we work over a field of characteristic zero. In this case the spectral sequence collapses. The tree space, Tn, which is used to describe Γ-homology has a natural action of the symmetric group Sn. We identify the representation of Sn on its only non-trivial homology group as that given by the first Eulerian idempotent en(l) in QSn. Using this, we prove that Γ-homology coincides with the existing theories over a field of characteristic zero. In fact, the tree space, Tn, gives a representation of Sn+l. In Chapter IV we calculate the character of this representation. Moreover, we show that each Eulerian representation of Sn is the restriction of a representation of Sn+1. These Eulerian representations are given by idempotents en(j), for j=1, ..., n, in QSn, and occur in the work of Barr [B], Gerstenhaber and Schack [G-S, 1], Loday [L, 1,2,3] and Hanlon [H]. They have been used to give decompositions of the Hochschild and cyclic homology of commutative algebras in characteristic zero. We describe our representations of Sn+1 as virtual representations, and give some partial results on their decompositions into irreducible components. In Chapter V we return to commutative algebra homology, now considered in prime characteristic. We give a corrected version of Gerstenhaber and Schack's [G-S, 2] decomposition of Hochschild homology in this setting, and give the analagous decomposition of cyclic homology. Finally, we give a counterexample to a conjecture of Barr, which states that a certain modification of Harrison cohomology should coincide with André/Quillen cohomology