Adjoint functors on the derived category of motives

Voevodsky’s derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper we 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. We 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, we determine exactly which adjoint functors exist. For a field k and commutative ring R, we consider three subcategories of the derived category of motives, DM(k;R): the category DMT (k;R) of mixed Tate motives, the category DMeff(k;R) of effective motives, and the category D0(k;R) of (non-effective) motives of dimension ≤ 0. Each is a localizing subcategory of DM(k;R), meaning a full triangulated subcategory that is closed under arbitrary direct sums in DM(k;R). It is a useful formal property of the category DM(k;R) that it contains the direct sum and the product of an arbitrary set of objects, not necessarily finite. In these three cases, Neeman’s Brown Representability Theorem [15] implies that the inclusion f∗ of the subcategory has a right adjoint f∗, and that f∗ in turn has a right adjoint f (1): f∗ a f∗ a f (1) For example, if f∗ denotes the inclusion of DMT (k;R) into DM(k;R), the existence of f∗ means that for every motive M in DM(k;R), there is a mixed Tate motive C(M) with a map C(M)→M that induces an isomorphism on motivic homology. This functor has been useful, for example in characterizing mixed Tate motives as the motives which satisfy the motivic Künneth property [25, Theorem 7.2]. The functor f (1) has probably not been considered before. On the other hand, for many fields k and rings R, and for the three subcategories mentioned above, the sequence of adjoint functors above cannot be extended to the left or right, because of various failures of finite generation for motivic cohomology. For example, for any algebraically closed field k which is not the algebraic closure of a finite field, we show that the inclusion f∗ of DMT (k;Q) into DM(k;Q) does not have a left adjoint, using that the Mordell-Weil group of an elliptic curve over k has infinite rank. In particular, it follows that a product of mixed Tate motives need not be mixed Tate. We deduce that the analogous subcategory of cellular spectra in the stable homotopy category SH(k) is not closed under products for some fields k. (The opposite conclusion has been announced at least once.) By results of Balmer, Dell’Ambrogio, and Sanders, in the case of DMT (k;R) (but not for the other subcategories we consider), f∗ has a left adjoint if and only