Adams Operations and the Dennis Trace Map

Abstract

For a commutative algebra A, the algebraic K-theory of A, K*(A), and the Hochschild homology of A, HH*(A), are graded rings, and the Dennis trace map D : K *(A) → HH*( A) is a graded ring map. Since Hochschild homology is a more user friendly theory than the algebraic K-theory, one would like to use the Dennis trace map to study the algebraic K-theory via Hochschild homology. For example, this idea was used by Geller and Weibel to give a counterexample to a conjecture of Beilinson and Soule on the vanishing of certain components of K*( A). To further study the algebraic K-theory via the Dennis trace map, one would like to know what additional structure the Dennis trace map preserves. In the first part of this thesis we prove a conjecture of Loday, Geller and Weibel that rationally, the Dennis trace map preserves the Adams operations and the Hodge decomposition. In the second part of the thesis we give a tool for comparing the Adams operations on K-theory with the ones on Hochschild homology in the non rational case. We do so by giving a formula for the Dennis trace map, as a map from a split version of the S-construction model for K-theory to additive cyclic nerve model of Hochschild homology. The motivation to find such a formula is Grayson's explicit description of the Adams operations on the S-construction for K-theory and McCarthy's explicit description of the Adams operations on the additive cyclic nerve complex for Hochschild homology.