Summary: THE SLICE FILTRATION ON DM(k) DOES NOT PRESERVE
In this appendix we give an unconditional argument for the following (un)-property
of the slice filtration on DM(k):
Proposition 0.1 -- The slice filtration on DM(k) does not preserve geometric
Recall that the slice filtration is a sequence of transformations:
· · · id
(M) = (M(-n))(n) with : DM(k) DMeff(k) the right adjoint to
the full embedding DMeff(k) DM(k). When M is effective (e.g.. the motive
M(X) of a smooth projective variety X) we have (M(-n)) = Homeff(Z(n), M)
where Homeff stands for the internal hom in DMeff(k). We will prove the following:
Proposition 0.2 -- Assume that k is big enough. There exists a smooth
projective k-variety X such that Homeff(Z(1), M(X)) is not a geometric motive.
We will implicitly assume k algebraically closed and work with rational coefficients.
1. Compacity in DMeff(k)