ADJOINT FUNCTORS ON THE DERIVED CATEGORY OF MOTIVES

Abstract

We show that the subcategory of mixed Tate motives in Voevodsky’s derived category of motives is not closed under infinite products. In fact, the infinite product$\prod _{n=1}^{\infty }\mathbf{Q}(0)$is not mixed Tate. More generally, the inclusions of several subcategories of motives do not have left or right adjoints. The proofs use the failure of finite generation for Chow groups in various contexts. In the positive direction, we show that for any scheme of finite type over a field whose motive is mixed Tate, the Chow groups are finitely generated.