Tom Bachmann (Mainz): cdh-local motivic cohomology for singular schemes

This talk was part of the workshop "Motivic and non-commutative aspects of enumerative geometry" and the conference "Homotopy theory, K-theory, and trace methods" held in July 2023 at Radboud University, Nijmegen, The Netherlands. The latter conference was held in honor of Bjørn Ian Dundas' 60th birthday.

Abstract: The formalism of the motivic stable categories allows us to define motivic cohomology of arbitrary schemes: just pull back the motivic spectrum HZ constructed by Spitzweck over the integers to any scheme, and evaluate. In this talk I will explain how the above very abstract definition can be identified with a more concrete one, and also reconciled with other recent approaches to motivic cohomology of general schemes via syntomic cohomology. After that I will mention two applications: the resolution of Voevodsky’s zero-slice conjecture as well as the construction of the motivic filtration on homotopy K-theory, both over arbitrary base schemes. (jt. work with Elden Elmanto and Matthew Morrow)