Etale K-theory and motivic cohomology
Two key features of algebraic K-theory are its failure to
satisfy \'etale descent, and its motivic filtration in terms of higher Chow groups in the case of smooth schemes over a field (but expected more generally). I will explain a description of \'etale K-theory, which is the universal approximation to K-theory that satisfies \'etale descent; this is joint work with Dustin Clausen. Moreover, following the recent work of Bhatt--Morrow--Scholze on topological cyclic homology, I will also explain a construction of (an analog of) the motivic filtration on \'etale K-theory (and \'etale motivic cohomology) for arbitrary schemes (work in progress with Bhargav Bhatt and Dustin Clausen).
The discussion for Akhil Mathew’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2020-11-06-am (and will be deleted after ~3-7 days).