Seminar des SFB/TRR 326 GAUS From p-adic Hodge Theory to Motivic Cohomology and Back

The initial goal of p-adic Hodge theory, as formulated by the foundational conjectures of Fontaine in the 1980s, is to compare the different p-adic cohomology theories one can associate to schemes of mixed characteristic (0,p). If Fontaine's conjectures have now been solved by the work of many people, the recent development of prismatic cohomology has shed new light on integral aspects of this theory. In this talk, I want to explain how one can use these recent advances in p-adic Hodge theory to construct a new theory of motivic cohomology for general (qcqs) schemes. This theory generalises the recent construction of Elmanto-Morrow over a field to mixed characteristic, and allows us to give a simplified motivic approach to certain classical results in p-adic Hodge theory. This is part of a joint work in progress with Arnab Kundu.