Chow groups and intersection products for tensor triangulated categories

Abstract

This thesis deals with a generalization of an important family of invariants from algebraic geometry (the Chow groups of an algebraic variety) to the setting of tensor triangulated categories. It is shown that these tensor triangular Chow groups recover the usual notion of Chow groups of an algebraic variety when one applies the construction to the bounded derived category of coherent sheaves on a non-singular algebraic variety. Furthermore, it is proved that tensor triangular Chow groups enjoy good formal properties (such as functoriality and localization sequences) and some examples for the stable module category from modular representation theory are computed. In the last chapter of the thesis an intersection product on tensor triangular Chow groups is constructed, for tensor triangulated categories which admit an algebraic model and satsify a certain K-theoretic regularity condition analogous to the Gersten conjecture for algebraic varieties. It is shown that this product recovers the usual intersection product on a non-singular algebraic variety.