Geometry and Physics Seminar

In recent years, the work of E.--Morrow, Bouis and Kelly--Saito has produced a theory of motivic cohomology of an arbitrary scheme, enjoying favorable properties as predicted by Beilinson. I want to survey the difference (cofiber) between motivic cohomology and its "étale analog," an integral theory which 1) is the étale sheafification for smooth schemes over Dedekind domains, 2) is ``Beilinson's theory" rationally and 3) is syntomic cohomology at the bad primes and 4) is etale cohomology away bad primes. I will explain how studying this cofiber is related to several problems about algebraic cycles, birational geometry and unramified cohomology. This talk contains separate joint projects with Matthew Morrow, Nick Addington, the team of Annala-Bouis-E.-Raifei-Shin and some of my own work in progress.