Purity of Motivic Cohomology in mixed-characteristic (Arnab Kundu, Bucarest)

Motivic cohomology is a cohomology theory that can be defined internally within Grothendieck's category of motives. Voevodsky developed this theory for smooth varieties, demonstrating its profound connections to algebraic cycles and algebraic K-theory. However, its behaviour in mixed-characteristic remains less well understood. Building on recent advancements by Bachmann, Elmanto, Morrow, and Bouis, in a joint work with Bouis, we demonstrate a purity result over deeply-ramified bases. I will also discuss an application of this result in p-adic Hodge theory.