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arXiv · 2412.06635

Motivic cohomology of mixed characteristic schemes

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

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Tess Bouis. 2026-09-07. Motivic cohomology of mixed characteristic schemes. https://arxiv.org/abs/2412.06635

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