Abstract

I will given an overview of some joint work with Elden Elmanto in which we construct a non-A^{1-invariant motivic cohomology theory for schemes over fields. It is built by glueing A}1-invariant, cdh-local, motivic cohomology (developed in a separate project joint with Tom Bachmann and Elmanto) to contributions from either syntomic cohomology or derived de Rham cohomology. It satisfies various properties akin to classical motivic cohomology, such as an Atiyah—Hirzebruch spectral sequence converging to algebraic K-theory, the projective bundle formula, some Beilinson—Lichtenbaum type formulae, and the degree 2d, weight d motivic cohomology is related to zero cycles on singular varieties. But there are also new phenomena motivated by the structure of algebraic K-theory; for example, it has a vanishing range which refines Weibel’s vanishing conjecture and satisfies pro-cdh descent.