2025/06/11 by Tess Bouis, Bouis, Tess, Arnab Kundu +1 · 2 citations
#math.AG #math.KT #math.NT
paper · pdf · doi:10.48550/arxiv.2506.09910
The goal of this paper is to study non-\mathbbA1-invariant motivic cohomology, recently defined by Elmanto, Morrow, and the first-named author, for smooth schemes over possibly non-discrete valuation rings. We establish that the cycle class map from p-adic motivic cohomology to a suitable truncation of Bhatt--Lurie's syntomic cohomology is an isomorphism, thereby verifying the Beilinson--Lichtenbaum conjecture in this generality. As a first consequence, we prove that this motivic cohomology integrally recovers the classical definition of motivic cohomology in terms of Bloch's cycle complexes, whenever the latter is defined. As a second consequence, we show a purity theorem for this cohomology theory over perfectoid rings, thus motivically refining a result of Nizioł in algebraic K-theory. The key ingredient in our approach is a version of Gabber's presentation lemma applicable in mixed characteristic, non-noetherian settings.