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.