Motivic cohomology of mixed characteristic schemes

TL;DR

This paper develops a new motivic cohomology theory for schemes in mixed characteristic, connecting it with topological cyclic homology, derived de Rham, and syntomic cohomology, extending previous constructions over fields.

Contribution

It introduces a non-$A^1$-invariant motivic cohomology for mixed characteristic schemes using a global filtration on topological cyclic homology, generalizing prior work over fields.

Findings

Provides a new integral refinement of derived de Rham cohomology.

Establishes relations to étale cohomology and algebraic K-theory.

Ensures the theory satisfies key motivic properties.

Abstract

We introduce a theory of motivic cohomology for quasi-compact quasi-separated schemes, which generalises the construction of Elmanto--Morrow in the case of schemes over a field. Our construction is non-$\mathbb{A}^1$-invariant in general, but it uses the classical $\mathbb{A}^1$-invariant motivic cohomology of smooth $\mathbb{Z}$-schemes as an input. The main new input of our construction is a global filtration on topological cyclic homology, whose graded pieces provide an integral refinement of derived de Rham cohomology and Bhatt--Morrow--Scholze's syntomic cohomology. Our theory satisfies various expected properties of motivic cohomology, including relations to \'etale cohomology and to non-connective algebraic $K$-theory.