Time: 10:00-11:30, 12:30-14:00, 11/24, 12/1, 12/8, 12/15, 12/22

This course aims to give an introduction to some recent progress on p-adic Hodge theory. The whole course is divided into two parts.

In the first part, I will give an introduction to Scholze’s work on p-adic Hodge theory for rigid analytic varieties; more precisely, I will explain Scholze’s pro-etale site, and the de Rham comparison theorem for proper and smooth analytic variety over a finite extension of Q_p.

The second part of the course will be devoted to prismatic cohomology, which was introduced by Bhatt—Scholze in 2019. Such a theory provides a uniform framework to compare various cohomology theories (crystalline cohomology, q-de Rham cohomology, p-adic etale cohomology…) for smooth p-adic formal schemes. After introducing the basic notions on  prismatic site, I will try to explain the fundamental results on prismatic cohomology: Hodge—Tate comparison, crystalline comparison and etale comparison theorems. 

Prerequisite: 

Commutative algebra, and homological algebra. In particular,  I will use freely the language of derived category and derived functor. Basic knowledge on Grothendieck topology and sheaf theory is essential for the understanding of various construction of the course (I will rapidly review this in the course). Finally, basic knowledge on perfectoid spaces is assumed in the first part. 

References: 

1. P. Scholze, p-adic Hodge theory for rigid analytic varieties, Forum of Mathematics, Pi (2013), Vol. 1, e1, 77 pages, doi:10.1017/fmp.2013.1,

2. B. Bhatt, P. Scholze, Prisms and prismatic cohomology, preprint in 2019,  https://arxiv.org/abs/1905.08229

3. M. Morrow, T. Tsuji, Generalised representations as q-connections in integral p-adic Hodge theory, preprint in 2020, https://arxiv.org/pdf/2010.04059.pdf

4. Kimihiko Li, Prismatic and q-crystalline sites of higher level, preprint in 2021, https://arxiv.org/pdf/2102.08151.pdf