Kelly S. Voevodsky Motives and ldh-descent

Paris: Société mathématique de France, 2017, — 134 p.Voevodsky defines a triangulated category /K(k) and shows that this category possesses many of the properties that the derived category of a
conjectural category of mixed motives would satisfy. One unsatisfactory aspect of his theory was that most of the results about non-smooth varieties, for example that higher Chow groups appear as hom groups in /K(k), were only available over a characteristic zero base field.
Working with Z[1/p]-coefficients, we obtain all the results of Friedlander, Suslin, and Voevodsky in [96] over a perfect field of exponential characteristic p.
The strategy is to replace Voevodsky’s application of resolution of singularities via the cdh-topology, with Gabber’s theorem on alterations, cf. [40], via a refinement of the cdh-topology which we christen the ldh-topology. As Gabber’s theorem only provides an analogue of the existence of a desingularisation, and not the weak factorisation theorem, the proof of cdh-descent of [19] cannot be adapted. We instead use a completely different strategy, proving cdh-descent using Ayoub’s proper base change theorem, [3], for the stable homotopy category of Morel and Voevodsky