Some recent advances in syntomic cohomology (3/3)
By Akhil Mathew
Bhatt-Morrow-Scholze have defined integral refinements $Z_p(i)$ of the syntomic cohomology of Fontaine-Messing and Kato. These objects arise as filtered Frobenius eigenspaces of absolute prismatic cohomology and should yield a theory of "p-adic étale motivic cohomology" -- for example, they are closely related to p-adic K-theory and topological cyclic homology. Moreover, they compare naturally to p-adic étale cohomology of the generic fiber. I will give an overview of some of the recent advances in the subject (due to many authors) and explain the identification (joint with Bhargav Bhatt) for regular p-torsionfree schemes.