Abstract
We study mixed versions of the classical quotient functor from Chow motives to numerical motives. We compare two natural definitions, which turn out to be very different. We investigate fullness, conservativity and exactness of these two functors.
Highlights
Keeping the above results in mind, in order to understand algebraic cycles over k-schemes that are smooth but not projective, it seems useful to provide a mixed analogue of such a functor, which would extend num from CHM(k) to DMgm(k), Voevodsky’s triangulated category of geometric motives
We argue as above to conclude that δa+1 ∈ N, and π = 0
Remark 6.3 (1) If one wants to work with a motivic category which is larger than T one could consider a similar construction due to Bondarko [5, Remark 6.3.2(3)], who defines a triangulated functor
Summary
Keeping the above results in mind, in order to understand algebraic cycles over k-schemes that are smooth but not projective, it seems useful to provide a mixed analogue of such a functor, which would extend num from CHM(k) to DMgm(k), Voevodsky’s triangulated category of geometric motives. Theorem 1.1 Let T ⊂ DMgm(k) be the smallest triangulated category containing finite dimensional Chow motives. When k is algebraic over a finite field, the functor num is conjectured to be an equivalence of categories.
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