Rigid Analtyic Geometry and Perfectoid Spaces

We will be covering the basics of Rigid Analytic Geometry in the first half of the seminar, then moving onto adic and perfectoid spaces. Our main source for the first half will be [B]. After this, we will likely follow [C].


September 14
Alex Scheffelin
What is Rigid Geometry and Tate Algebras (Chapters 1 & 2 of [B])
We will discuss what rigid geometry is, and begin by introducing the notion of a Tate algebra, a fundamental notion in rigid geometry.
September 21
Alex Scheffelin
Finishing Tate Algebras (Chapter 1 of [B])
We will finish our exposition of chapter 2, completing proofs of several elementary facts about Tate algebras, and then proving that ideals are complete, and thus closed.
Notes for 14th and 21st: notes
September 28
Kevin Chang
Affinoid Algebras and Affinoid Spaces (Sections 3.1 & 3.2 of [B])
We will introduce an affinoid algebra, which is the rigid analog of a finite type K-algebra. We also introduce Sp A, the affinoid K-space associated to A which plays a role analogous to Spec in algebraic geometry.
October 5
Benjamin Church
Affinoid Subdomains (Section 3.3 of [B])
The Zariski topology on an affinoid K-space is coarse. We will introduce a finer topology which is induced directly from K's topology.
October 12
Alex Scheffelin
Affinoid Functions (Sections 4.1 & 4.2 of [B])
We first define germs of affinoid functions, which is a construction very similar to the local rings for schemes. Following this, we examine locally closed immersions of affinoid spaces, providing a local way to describe affinoid subdomains.
October 19
Alex Scheffelin
Tate's Acyclicity Theorem (Sections 4.3 of [B])
Due to the topology on affinoid spaces, the sheaf condition for the presheaf of affinoid functions is rarely satisfied. The presheaf will always be separated, but the ability to glue functions will fail. We will be able to prove that given a finite cover, we will be able to glue functions, which is a special case of Tate's Acyclicity Theorem which we will prove.
October 25
Alex Scheffelin
Motivations for Adic Spaces (Lecture 1 (Sept. 26) of [C])
We will explore a way to define sheaves on the closed points of nice schemes (say for varieties) in a way that is equivalent to defining them on the scheme itself. As a result we will find a way to characterize stalks of the sheaf at non-closed points without any reference to the point itself using the notion of a prime filter.
November 2
Alex Scheffelin
Spaces of Valuations (Lecture 2 (Oct. 3) of [B])
We will ultimately be basing adic spaces, which are spaces formed from valuations. We will review some of the basics of valuation theory, then define the Riemann-Zariski space, a space of classical interest. Zariski using these was able to prove the resolution of algebraic varieties in dimensions 2 and 3, and even recently they have been used in proofs. We then finish by defining the valuation spectrum of a ring.
November 9
Amal Mattoo
Huber Rings (Lecture 5 (Oct. 24) of [B])
We introduce the notion of a Huber ring, a certain class of topological rings which will be of importance for us. We introduce the notion of boundedness, and prove some facts about Huber rings and their topology.
November 16
Morena Porzio
Generalities on Constructible Sets and Spectral Spaces (Lecture 3 (Oct. 10) of [B])
In general, perfectoid spaces will not enjoy any kind of Noetherianity condition. However, a motoo to keep in mind is that several "finiteness questions" can find an answer by dealing with the constructible topology. Therefore we want to analyze quasi-compactness properties, which will lead us to a digression about (pro-)constructible sets and spectral spaces. We will then discuss some characterizations of constructibility, such as Hochster's criterion. We will end by sketching that Spv(A) is spectral.
November 23
We enjoyed a nice lunch at my apartment to celebrate Thanksgiving.
November 30
Alex Scheffelin
Specialization Constructions (Lecture 4 (Oct. 17) of [B])
We will examine the topology of Spv(A), and will consider the specialization relation.
December 7
Alex Scheffelin
Huber Rings Part II (Lecture 6 (Oct. 31) of [B])
We have to further develop the theory of Huber rings in order to have a hope of working with adic spaces. Towards this end we will discuss topologically nilpotent units, and how to topologize a localization.