I’ve posted a new paper here (and on the arXiv). Voevodsky’s derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper I study whether the inclusions of three important subcategories of motives have a left or right adjoint. These adjoint functors are useful constructions when they exist, describing the best approximation to an arbitrary motive by a motive in a given subcategory. I find a fairly complete picture: some adjoint functors exist, including a few which were previously unexplored, while others do not exist because of the failure of finite generation for Chow groups in various situations. For some base fields, I can determine exactly which adjoint functors exist.
Drawing by Robert Leighton, from The New Yorker