Motivic cohomology of mixed characteristic schemes

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.