Doctor of Philosophy (PhD)
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. Except for the case p=1, which yields an algebraic group, the Chow groups remain mysterious. This thesis 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. Bloch's formula follows from the fact that the coniveau spectral sequence for algebraic K-theory on X induces flasque resolutions of these sheaves. Algebraic cycles and algebraic K-theory are thereby related via the general method of coniveau filtration of a topological space. Hence, "linearization" of the Chow groups is 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, which I will call K-theory, augmented K-theory, relative K-theory, and relative negative cyclic homology. The objects of principal interest are certain sheaf cohomology groups defined in terms of these theories. In particular, the pth Chow group is the pth cohomology group of the pth K-theory sheaf. Due to the critical role of coniveau filtration, I refer to this construction as the coniveau machine. The main theorem in this thesis establishes the existence of the coniveau machine for algebraic K-theory on a smooth algebraic variety. This result depends on a large body of prior work of Bloch-Ogus, Thomason, Colliot-Thelene, Hoober, and Kahn, Loday, and most recently, Cortinas, Haesemeyer, and Weibel. An immediate corollary is a new formula expressing generalized tangent groups of Chow groups in terms of negative cyclic homology, which is more tractable than algebraic K-theory.
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Dribus, Benjamin F., "On the infinitesimal theory of Chow groups" (2014). LSU Doctoral Dissertations. 821.