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Perfectoid Spaces MATH 679: PERFECTOID SPACES BHARGAV BHATT∗ Course Description. Perfectoid spaces are a class of spaces in arithmetic geometry introduced in 2012 by Peter Scholze in his PhD thesis [Sch12]. Despite their youth, these spaces have had stunning applications to many different areas of mathematics, including number theory, algebraic geometry, representation theory, and commutative algebra. The key to this success is that perfectoid spaces provide a functorial procedure to translate certain algebro-geometric problems from characteristic 0 (or mixed characteristic) to characteristic p; the latter can often be more accessible thanks to the magic of Frobenius. A major portion of this class will be devoted to setting up the basic theory of perfectoid spaces. En route, we will encounter Huber's approach to nonarchimedean geometry via his language of adic spaces, Faltings' theory of `almost mathematics' conceived in his proof of Fontaine's conjectures in p-adic Hodge theory, and the basic algebraic geometry of perfect schemes in characteristic p. The highlight of this part of the course will be the `almost purity theorem', a cruder version of which forms the cornerstone of Faltings' aforementioned work. The rest of the course will focus on a single application of perfectoid spaces. There are several choices here: we could go either in the arithmetic direction (such as Scholze's work on the weight-monodromy conjecture or some recent progress in p-adic Hodge theory) or in a more algebraic direction (which is the lecturer's inclination). A final choice will be made during the semester depending on audience makeup and interest. Contents 1. January 4th........................................................................................ 3 1.1. Overview...................................................................................... 3 1.2. Perfectoid fields................................................................................ 3 1.3. Perfectoid spaces............................................................................... 5 2. January 9th........................................................................................ 6 2.1. Overview (continued).......................................................................... 6 2.2. Conventions................................................................................... 7 2.3. Perfections and tilting......................................................................... 8 3. January 11th...................................................................................... 9 3.1. Perfectoid fields................................................................................ 9 4. January 18th...................................................................................... 11 4.1. Perfectoid fields (continued).................................................................... 11 5. January 23rd...................................................................................... 14 5.1. Almost mathematics........................................................................... 14 6. January 25th...................................................................................... 18 6.1. Almost mathematics (continued)............................................................... 18 6.2. Almost ´etaleextensions........................................................................ 20 7. January 30th...................................................................................... 20 7.1. Almost ´etaleextensions (continued)............................................................ 20 7.2. Construction (finite ´etalealgebras and idempotents)............................................ 21 7.3. Almost purity in characteristic p ............................................................... 22 ∗Notes were taken by Matt Stevenson, who is responsible for any and all errors. Many thanks to Takumi Murayama and Emanuel Reinecke for sending corrections. Please email [email protected] with any corrections. Compiled on April 27, 2017. 1 2 BHARGAV BHATT 8. February 1st....................................................................................... 23 8.1. Almost purity in characteristic p (continued)................................................... 23 8.2. Non-Archimedean Banach algebras (algebraically).............................................. 26 9. February 6th....................................................................................... 27 9.1. Non-Archimedean Banach algebras (continued)................................................. 27 9.2. Perfectoid algebras............................................................................. 29 10. February 8th...................................................................................... 29 10.1. Perfectoid algebras (continued)............................................................... 30 11. February 13th.................................................................................... 32 11.1. Cotangent complexes......................................................................... 33 12. February 15th.................................................................................... 36 12.1. Cotangent complexes (continued)............................................................. 36 12.2. Tilting by Fontaine's functors................................................................. 38 13. February 20th.................................................................................... 39 13.1. Almost purity................................................................................. 40 13.2. Adic spaces................................................................................... 41 14. February 22nd.................................................................................... 42 14.1. Basic properties of Tate rings................................................................. 42 14.2. Affinoid (pre-)adic spaces..................................................................... 43 15. March 6th........................................................................................ 45 15.1. Affinoid adic spaces (continued)............................................................... 46 15.2. Microbial valuation rings...................................................................... 47 16. March 8th........................................................................................ 49 16.1. Properties of affinoid adic spaces.............................................................. 49 16.2. Global structure of affinoid adic spaces........................................................ 51 17. March 20th....................................................................................... 52 17.1. Global structure of affinoid adic spaces (continued)............................................ 53 17.2. The structure presheaf........................................................................ 54 18. March 22nd....................................................................................... 55 18.1. The structure presheaf (continued)............................................................ 55 18.2. Background on henselian pairs................................................................ 57 19. March 27th....................................................................................... 58 19.1. Adic spaces................................................................................... 60 19.2. Analytic aspects of perfectoid spaces.......................................................... 61 20. March 29th....................................................................................... 62 20.1. Affinoid perfectoid spaces..................................................................... 63 21. April 3rd......................................................................................... 66 21.1. Affinoid perfectoid spaces (continued)......................................................... 66 22. April 5th......................................................................................... 69 22.1. Affinoid perfectoid spaces (continued)......................................................... 69 23. April 10th........................................................................................ 71 23.1. Tate acyclicity................................................................................ 71 24. April 12th........................................................................................ 74 24.1. Tate acyclicity (continued).................................................................... 74 24.2. \Noetherian approximation" for perfectoid algebras........................................... 76 25. April 17th........................................................................................ 77 25.1. Tate acyclicity (continued).................................................................... 77 25.2. Perfectoid spaces............................................................................. 78 MATH 679: PERFECTOID SPACES 3 25.3. The ´etaletopology............................................................................ 80 26. April 19th........................................................................................ 81 26.1. The ´etaletopology (continued)................................................................ 81 26.2. Proof of the Almost Purity Theorem.......................................................... 83 27. April 25th∗ ......................................................................................
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