$\mathbb{A}^1$-invariant motivic cohomology of schemes

Abstract

Voevodsky outlined a conjectural programme that his slice filtration in motivic homotopy theory should give rise to a good theory of $\mathbb{A}^1$-invariant motivic cohomology. This paper achieves his vision in the generality of arbitrary quasicompact, quasiseparated schemes, by introducing a theory of $\mathbb{A}^1$-invariant motivic cohomology which is related to Weibel's homotopy $K$-theory via an Atiyah--Hirzebruch spectral sequence, and which we compare to étale and syntomic cohomology in the style of the original conjectures of Beilinson and Lichtenbaum. In addition, it is represented by an absolute motivic spectrum and therefore satisfies cdh descent, and modules over it offer a candidate for the derived category of $\mathbb{A}^1$-invariant motives. We establish some of Voevodsky's open conjectures on slices, in particular relating the zeroth slice of the motivic sphere to homotopy $K$-theory. In the final section we prove analogous results for the Hermitian $K$-theory of qcqs schemes on which $2$ is invertible. As an auxiliary tool we introduce cdh-motivic cohomology, defined as the cdh sheafification of the left Kan extension of the motivic cohomology of smooth $\mathbb{Z}$-schemes. We offer a new approach to control the latter, independent of previous work on $\mathbb{A}^1$-invariant motivic cohomology of smooth schemes over mixed characteristic Dedekind domains: our approach is based on recent developments in $p$-adic cohomology, in particular syntomic and prismatic cohomology. The cdh-motivic cohomology is also a necessary ingredient in the last two authors' and Bouis' construction of non-$\mathbb{A}^1$-invariant motivic cohomology of qcqs schemes.