Beilinson--Lichtenbaum phenomenon for motivic cohomology

Abstract

The goal of this paper is to study non-$\mathbb{A}^1$-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 Nizio\l{} 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.