Perfectoid Spaces

Following up on the seminar last semester we will change pace and work on attempting to understand a use of perfectoid spaces / rings in proofs, and then work to understand the machinery needed for the proof.

• When: 6:10-7:40 on Thursdays
• Where: Math 528s
• References:

  [C] Conrad Number theory learning seminar 2014-2015

  [A] André La conjecture du facteur direct. This is André's original proof of the direct summand conjecture.

  [B1] Bhatt On the direct summand conjecture and its derived variant. Bhatt's paper which also proves the direct summand conjecture and a derived version conjectured by de Jong.

  [B2] Bhatt On the direct summand conjecture and its derived variant. Bhatt's paper which also proves the direct summand conjecture and a derived version conjectured by de Jong.

  [BMS] Bhatt, Morrow, Scholze Integral p-adic Hodge theory. A reference for Integral perfectoid rings, specifically section 3.

  [BIM] Bhatt, Iyengar, Ma Regular rings and perfect(oid) algebras. It presents a p-adic analogue of Kunz's theorem which says that a Noetherian ring of characteristic \(p\) is regular if and only if the Frobenius map is flat.

  [SW] Scholze, Weinstein Berkeley Lectures on \(p\)-adic Geometry. A rapid introduction to adic spaces, perfectoid rings/spaces, etc.

• Any notes taken will be added here.

Schedule

January 25

Alex Scheffelin

Recap
We discussed how we want to proceed, and came up with a rough plan for what we will cover. We discussed some classical theorems such as the Cohen Structure Theorem and Kunz's theorem, and discussed a \(p\)-adic variant of Kunz's theorem which will be the next topic we discuss.

 

Alex Scheffelin

2/2

\(p\)-adic Kunz's Theorem ([BIM])

I will discuss the contents of the paper "Regular rings and perfect(oid) algebras" which presents a \(p\)-adic analogue of Kunz's theorem which says that a characteristic \(p\) Noetherian ring is regular if and only if the Frobenius map is flat. This paper proves (a slightly stronger result even) that a \(p\)-adically complete Noetherian ring is regular if and only if it admits a faithfully flat map to a (integral) perfectoid ring. In characteristic \(p\) we can take the perfection of our ring as the perfectoid ring, and Kunz's theorem says that the map is faithfully flat. We will go over the general strategy of the proof, with a particular focus on the properties of perfectoid rings needed to prove the result, which we will then spend the next lecture trying to establish.

 

Alex Scheffelin

2/9

Witt Vectors
I will speak about Witt Vectors, defining them, a few endomorphisms of them, and their most relevant properties. I will mostly be following the treatment in Bourbaki.

 

Alex Scheffelin

2/16

Properties of perfectoid rings continued (BMS)
We will finish establishing some results on perfectoid rings.

 

Alex Scheffelin

2/23

Properties of perfectoid rings continued continued (BMS)
We will actually finish establishing some results on perfectoid rings.

 

Caleb Ji

3/2

Weight Monodromy
We discussed the statement of weight monodromy.

 

Cancelled

3/9

Cancelled
Due to illness we cancelled.

 

None

3/16

Spring Break
We are on Spring Break.

 

Alex Scheffelin

3/23

Motivating Adic Spaces (Lecture 2 of [SW])
We will motivate the definition of adic spaces as a concept unifying formal schemes and rigid analytic varieties. We then review some algebra which is necessary to define adic spaces.

 

Caleb Ji

3/30

Defining Adic Spaces (Lecture 3 of [SW])
We will define adic spaces. An issue arises where the structure "sheaf" may only be a presheaf, but in some particular cases on interest this is not the case. Some cases where this is not the case include: schemes, formal schemes, rigid spaces which are our main objects of interest. However, we also define a notion of a pre-adic space which resembles the definition of an algebraic space by enlarging the category of schemes inside of the functor category.

 

Alex Scheffelin

4/6

Examples of Adic Spaces and Analytic Points. (Lecture 4 of [SW])
We will see various examples of adic spaces. We will define a notion of an "analytic point" and analyze the adic spectrum \(\mathrm{Spa}\mathbb{Z}_p[[T]]\), the adic open unit disk over \(\mathbb{Z}_p\).

 

Alex Scheffelin

4/13

Examples of Adic Spaces and Analytic Points Continued. (Lecture 4 of [SW])
We will see various examples of adic spaces. We will define a notion of an "analytic point" and analyze the adic spectrum \(\mathrm{Spa}\mathbb{Z}_p[[T]]\), the adic open unit disk over \(\mathbb{Z}_p\).

 

Amal Mattoo

4/20

Perfectoid Rings. (Lecture 6 of [SW])
We will define a perfectoid ring which forms the basis for perfectoid spaces. These are different from integral perfectoid rings as we defined earlier as a perfectoid ring in this context is necessarily Tate.

 

Amal Mattoo

4/20

Perfectoid Spaces. (Lecture 7 of [SW])
We define perfectoid spaces!