 
Summary: THE SLICE FILTRATION ON DM(k) DOES NOT PRESERVE
GEOMETRIC MOTIVES
JOSEPH AYOUB
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
motives.
Recall that the slice filtration is a sequence of transformations:
n
n1
· · · id
where n
(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 kvariety 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)
