On the Infinitesimal Theory of Chow Groups

Abstract

The Chow groups of codimension-p algebraic cycles modulo rational equivalence on a smooth algebraic variety X have steadfastly resisted the efforts of algebraic geometers to fathom their structure. This book explores a "linearization" approach to this problem, focusing on the infinitesimal structure of the Chow groups near their identity elements. This method was adumbrated in recent work of Mark Green and Phillip Griffiths. Similar topics have been explored by Bloch, Stienstra, Hesselholt, Van der Kallen, and others. A famous formula of Bloch expresses the Chow groups as Zariski sheaf cohomology groups of algebraic K-theory sheaves on X. "Linearization" of the Chow groups is thereby related to "linearization" of algebraic K-theory, which may be described in terms of negative cyclic homology. The "proper formal construction" arising from this approach is a "machine" involving the coniveau spectral sequences arising from four different generalized cohomology theories on X, with the last two sequences connected by the algebraic Chern character. Due to the critical role of the coniveau filtration, I refer to this construction as the coniveau machine. The main theorem in this book establishes the existence of the coniveau machine for algebraic K-theory on a smooth algebraic variety. An immediate corollary is a new formula expressing generalized tangent groups of Chow groups in terms of negative cyclic homology.