Abstract:

The theory of prismatic cohomology was introduced by Bhatt—Scholze and has lots of applications to number theory and arithmetic geometry. A stacky way to prismatic theory was very recently developed and studied by Drinfeld and Bhatt—Lurie. In this mini-course, we want to study a special stack, the Hodge-Tate stack, in the theory Bhatt—Lurie, and exhibit how the stack is related to the Sen theory (and p-adic non-abelian Hodge theory, if time permits).

We will recall some basic definitions in prismatic theory and study the Hodge—Tate crystals, following the works of Yichao Tian and Min-Wang, as the first impress. Then we will define and study the Hodge-Tate stack and show how to generalise the works on crystals to the derived case. If time permits, we will explain how to use the Hodge-Tate stack to study p-adic non abelian Hodge theory.

Reference:

[1] J. Anschutz, B. Heuer, A.-C. Le Bras: v-vector bundles on p-adic fields and Sen theory via the Hodge-Tate stack, arXiv:2211.08470

[2] J. Anschutz, B. Heuer, A.-C. Le Bras: Hodge-Tate stacks and non-abelian p-adic Hodge theory of v-perfect complexes on rigid spaces, arXiv:2302.12747

[3] B. Bhatt, P. Scholze: Prisms and prismatic cohomology, arXiv:1905.08229

[4] B. Bhatt, J. Lurie: Absolute prismatic cohomology, arXiv:2201.06120

[5] Y. Min, Y. Wang: On the Hodge-Tate crystals over O_K, arXiv:2112.10140

[6] Y. Tian: Finiteness and duality for the cohomology of prismatic crystals, arXiv:2109.00801

10/20  10/27  11/3  11/10  11/17  11/24  12/1  12/8  

13:00--15:00