Beilinson--Lichtenbaum phenomenon for motivic cohomology

TL;DR

This paper proves the Beilinson--Lichtenbaum conjecture for non-$A^1$-invariant motivic cohomology over various rings, establishing isomorphisms with syntomic cohomology and confirming classical recoverability.

Contribution

It extends the Beilinson--Lichtenbaum phenomenon to non-$A^1$-invariant motivic cohomology over non-discrete valuation rings and perfectoid rings, using a new Gabber presentation lemma.

Findings

Cycle class map is an isomorphism to syntomic cohomology.

Motivic cohomology recovers classical definitions over Bloch's cycle complexes.

Cohomology is $A^1$-invariant over perfectoid rings.

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 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. Over perfectoid rings, we show that this cohomology theory is actually $\mathbb{A}^1$-invariant, thus partially answering a question of Antieau--Mathew--Morrow. The key ingredient in our approach is a version of Gabber's presentation lemma applicable in mixed characteristic, non-noetherian settings.