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# Collection Akhil Mathew - Some recent advances in syntomic cohomology

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.

Appears in collection : Franco-Asian Summer School on Arithmetic Geometry

Organizer(s) Ahmed Abbes (CNRS & IHÉS), Ana Caraiani (Imperial College London ), Ariane Mézard (Sorbonne Université), Takeshi Saito (University of Tokyo), Takeshi Tsuji (The University of Tokyo), Daxin Xu (Chinese Academy of Sciences), Weizhe Zheng (Chinese Academy of Sciences).
Date(s) 6/2/22 - 6/3/22
• 01:04:16
published on June 20, 2022

## Some recent advances in syntomic cohomology (1/3)

By Akhil Mathew

01:01:04
published on June 20, 2022

## Some recent advances in syntomic cohomology (2/3)

By Akhil Mathew

01:02:26
published on June 20, 2022

## Some recent advances in syntomic cohomology (3/3)

By Akhil Mathew

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