Special Year Seminar
Motivic Cohomology of Mixed Characteristic Schemes
I will present a new theory of motivic cohomology for general (qcqs) schemes. It is related to non-connective algebraic K-theory via an Atiyah-Hirzebruch spectral sequence. In particular, it is non-$A^1$-invariant in general, but it recovers classical motivic cohomology on smooth schemes over a Dedekind domain after $A^1$-localisation. The construction relies on the syntomic cohomology of Bhatt-Morrow-Scholze and the cdh-local motivic cohomology of Bachmann-Elmanto-Morrow, and generalises the construction of Elmanto-Morrow in the case of schemes over a field.