Motivic complexes and the K-theory of automorphisms

It has long been suspected that the conjectural motivic cohomology groups of a smooth variety X are related to the algebraic K-groups of X. Explicitly, it is expected that there is a spectral sequence whose terms are the motivic cohomology groups of X which converges to the algebraic K-groups of X. In fact, this expectation has led to proposed definitions of the motivic cohomology groups.In this thesis, we study a spectral sequence defined recently by Grayson that is conjectured to fill such a role. The $E\sb1$-terms of Grayson's spectral sequence are given by the homology groups of a family of explicit chain complex involving the Grothendieck groups associated to the category of projective modules of a ring equipped with a t-tuple of commuting automorphisms. The goal of this thesis is to study the $E\sb1$-terms of Grayson's spectral sequence, which are likely candidates for the motivic cohomology groups of X, and relate them with other proposed definitions of the the motivic cohomology groups. Specifically, we establish certain connections between Grayson's definition and a recently proposed definition of Voevodsky. The two definitions are not known to coincide in general, but we reduce the issue of their agreement to the case when X is the spectrum of a field. Further, we establish that Grayson's and Voevodsky's definitions are the same over finite coefficients when X is the spectrum of an algebraically closed field.U of I OnlyETDs are only available to UIUC Users without author permissio