Revisiting the Motivic Cohomology of Schemes

Since the original conjectures of Beilinson and Lichtenbaum in the 80s, several versions of motivic cohomology have been introduced and developed, notably by Voevodsky. Most classically, Bloch's higher Chow groups provide the accepted theory of motivic cohomology for smooth algebraic varieties, simultaneously related to algebraic cycles, étale cohomology, and algebraic K-theory. Secondly, Morel-Voevodsky's motivic homotopy theory yields a version of motivic cohomology for general schemes; but it is A1-invariant, so more closely related to the A1-invariant version of K-theory rather than to K-theory itself.

In this talk I will present some results from two projects. In the first, joint with Bachman and Elmanto, we use prismatic and syntomic cohomology to show that A1-invariant motivic cohomology is often controlled by smooth schemes in a simpler way than previously known. We use this new point of view to resolve several conjectures of Voevodsky from 2000. In the second project, joint with Elmanto, we prove in equal characteristic the existence of a natural ``A1-delocalisation'' of the A1-invariant theory, thereby constructing a new theory of motivic cohomology for arbitrary schemes over fields which is related to algebraic K-theory itself. Time permitting, I may discuss some applications to algebraic cycles on singular varieties.