Motivic cohomology of mixed characteristic schemes

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 étale cohomology and to non-connective algebraic $K$-theory.