Triangulated categories of motives in positive characteristic

TL;DR

This thesis extends Voevodsky's work on motives in positive characteristic by introducing the ldh topology, enabling unconditional theorems without resolution of singularities, and applies it to longstanding conjectures and comparisons in algebraic geometry.

Contribution

It develops the ldh topology and the concept of objects with traces, allowing for new descent results and removing the need for resolution of singularities in motive-related theorems.

Findings

Unconditional versions of Voevodsky's theorems in positive characteristic.

Comparison of cdh and ldh sheafifications and cohomologies.

Partial resolution of Weibel's 1980 conjecture.

Abstract

This thesis presents a way to apply this theorem of Gabber to a large portion of Voevodsky's work in order to lift the assumption that resolution of singularities holds. This gives unconditional versions of many of his and others' theorems provided we work Z[1/p] linearly, where p is the exponential characteristic of the base field. One example of the many applications we give is a partial answer to a 1980 conjecture of Weibel. Another is the removal of the hypothesis of resolution of singularities from a result of Suslin that compares Bloch's higher Chow groups and etale cohomology. Voevodsky's main tool in applying resolution of singularities is the cdh topology. We enlarge it slightly in order to apply this theorem of Gabber, presenting in this thesis a topology that we name the ldh topology, where l is a prime. We compare the cdh and ldh topologies using the concept of a "presheaf with traces", providing conditions under which the cdh and ldh sheafifications of a presheaf agree, as well as its cdh and ldh cohomologies. As far as applying resolution of singularities to motives goes, Voevodsky's most important theorem can be rephrased as a cdh descent condition, and we are led to ask for conditions under which certain objects in the Morel-Voevodsky stable homotopy category satisfy ldh descent. In order to compare cdh and ldh descent, we generalise the notion of a "presheaf with traces" to the concept of an "object with traces". We build on some results of Pelaez on the functoriality of the slice filtration to show that this concept of an "object with traces" interacts well enough with the slice filtration to provide the ldh descent that we need.