Ask about devissage arguments for vanishing and finiteness of motivic cohomology.
Asked in Toulouse, see my notes: Non-homotopy invariant categories? Example: understand first etale cohomology in “equal” characteristic.
What is expected in terms of comparison w chow cplxes or their hypercohom?
Try maybe first to for example prove the Parshin conjecture for varieties which are smooth projective and which sit as a fiber in a smooth projective Z-scheme.
Check that a compact object in DM_B is rigid. Which schemes over Z give rise to compact objects under f_! f^! ?
Ask for a private course on descent, survey of all topologies and facts such as Zariski and proper descent implies etale descent, does sheafification preserve descent properties etc etc etc.
nLab page on Questions intended for Cisinski