Seminar on perfectoid spaces

The student seminar meets Tuesdays and Thursdays at 1:30-3:00 pm in Math 731 during the Purdue summer session 2019. As of now, the seminar continues in the Fall 2019 semester on Tuesdays at 1:30-3:00 pm in Math 731.

The goal is to learn basic results about perfectoid spaces as well as the related machinery, such as adic spaces and almost mathematics.

References

B. Bhatt, lecture notes on perfectoid spaces - primary reference for the seminar.
P. Scholze, Perfectoid spaces - Scholze's thesis that developed perfectoid spaces.
M. Morrow, Adic and perfectoid spaces - a more general treatment of perfectoid rings/perfectoid spaces.
T. Wedhorn, Lecture notes on adic spaces.
S. Morel, Adic spaces.
P. Scholze, J. Weinstein, Berkeley lectures on p-adic geometry.
O. Gabber, L. Ramero, Almost ring theory.

Program

1.	Jun 11	(P. Čoupek) Complete nonarchimedean fields, direct and inverse limit perfections, Fontaine's tilting functor.
2.	Jun 13	(P. Čoupek) Tilting of p-adically complete rings, domains and valuation rings. Perfectoid fields.
3.	Jun 18	(P. Čoupek) Tilting of perfectoid fields to perfectoid fields.
4.	Jun 20	(P. Čoupek) Continuous valuations, preservation of continuous valuations under tilting of a perfectoid field.
5.	Jun 25	(P. Čoupek) Baby almost purity theorem (alg. closed fields), torsion-free modules over a valuation ring are flat, setup of almost mathematics.
6.	Jun 27	(P. Čoupek) Gabriel localization aka Serre quotient, realizing the category of almost modules as a full subcategory of modules.
7.	Jul 2	(J. Knight) Injectivity in the category of almost modules, almost flatness, almost projectivity and almost finite generation/presentation.
8.	Jul 9	(J. Knight) Almost étale algebras.
9.	Jul 16	(J. Knight) Almost purity theorem in characteristic p.
10.	Jul 18	(W. Sokurski) Nonarchimedean Banach K-algebras, their completions and various integral closures, description in terms of subrings of power-bounderd elements.
11.	Jul 23	(W. Sokurski) Cotangent complex.
12.	Jul 25	(P. Čoupek) Witt vectors, strict p-rings and their equivalence to perfect rings of char. p.
13.	Jul 30	(P. Čoupek) Integral perfectoid rings, A_inf and Fontaine's theta map.
14.	Aug 1	(P. Čoupek) Tilting correspondence for integral perfectoid rings and Tate perfectoid rings.
15.	Aug 6	(W. Sokurski) Cotangent complex and deformation theory.
16.	Aug 8	(W. Sokurski) Cotangent complex and deformation theory (cont.).
17.	Aug 20	(J. Knight) Almost purity theorem for fields.
18.	Sep 3	(J. Knight) Almost purity theorem for fields II.
19.	Sep 17	(P. Čoupek) Huber and Tate rings and pairs, bounded subrings, power bounded elements.
20.	Sep 24	(P. Čoupek) Affinoid fields, "classical" Tate algebras, perfectoid Tate algebras.
21.	Oct 15	(P. Čoupek) Adic spectrum, computation for affinoid fields.
22.	Oct 22	(P. Čoupek) Completion of Huber pairs preserves adic spectrum.
23.	Oct 29	(P. Čoupek) Localization of Huber pairs and structure presheaf of affinoids.
24.	Nov 5	(P. Čoupek) Criteria for sheafiness, standard and Laurent rational covers.