Modules of finite length and finite projective dimension, and algebraic cycles

In this talk, I will first interpret of an example of Dutta, Hochster and
McLaughlin as a failure of compatibility of the filtration by codimension of
support with the gamma filtration. I will then discuss joint work with
Roberts, using Thomason localisation in K-theory to relate this example, and
generalizations, to properties of algebraic cycles, including aspects of the
Standard Conjectures and the Bloch-Beilinson conjectures.