Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-theory

We outline the theory of motivic cohomology of general equicharacteristic schemes, as developed jointly with Matthew Morrow. Roughly, the lectures will be divided as follows: Lecture 1: cdh and A^1-invariant motivic cohomology. I will first give a general, leisurely introduction to the cdh topology and some of its applications to algebraic geometry and K-theory. Lecture 2: the construction of the motivic filtration on K-theory. I will then explain how to construct a motivic filtration on K-theory by gluing together the theory of syntomic cohomology and A^1-invariant/cdh motivic cohomology. Some of the results presented here are joint with Tom Bachmann and Matthew Morrow. Lecture 3: a sampler of motivic cohomology. I will then give some features of the resulting theory of motivic cohomology. Topics include an extension of the Nesterenko-Suslin isomorphism (with Milnor K-theory), a motivic refinement of Weibel's vanishing conjecture, and results on zero cycles.