The goal of this learning seminar is to study basics of rigid analytic geometry, and at the same time to prepare for the topic course given by Yiwen Ding in the Spring semester on perfectoid spaces. The pre-requisite is Chapters 2-3 of Hartshorne’s algebraic geometry. Some basic familiarity with $p$-adic numbers is needed. Each lecture will be two hours long. The main reference we propose is Part I of Siegfried Bosch’s book Lectures on Formal and Rigid Geometry. This reference is wonderful in that the logic flow is very clear, but the drawback is the lack of concrete examples and philosophical explanations. We recommend frequent consultation of Kedlaya’s lecture notes for intuitions and exercises.
(12/23) Shanxiao Huang: Introduction and organization. (notes)
Introduction to the topics of rigid analytic geometry and why we study it.
(1/3) Han Hu: Tate algebra I. (TeXed notes, speaker’s handout)
Cover [Bos] P9-P20 (until corollary 13). Discuss the definition of restricted power series $T_n$ and relevant properties. In particular, prove the maximum principle (Prop. 5), and then prove the theorem of Weierstraß Division (Thm. 8) and discuss its several corollaries (Cor. 9-13).
(1/5) Wenxuan Qi: Tate Algebra II. (TeXed notes)
Cover [Bos] P20-P29. Finish section 2.2 by deriving some more standard properties of $T_n$. Then prove Cor. 7 to show that each ideal of $T_n$ is finite and complete (Cor. 8) and generalize this results to finite generated $T_n$-modules (Cor. 10).
(1/9) Yiming Tang: Affinoid algebras and affinoid spaces. (TeXed notes, speaker’s handout)
Cover [Bos] Sect. 3.1, 3.2. State some immediate consequences from last chapter for affinoid algebras (Prop. 3-5). Then discuss the residue norm and the supremum norm and their relations, in particular, prove Thm. 17 and show that all residue norms are equivalent (Prop. 20). Finally, introduce the affinoid spaces (Sect. 3.2).
(1/12) Wenhan Dai: Affinoid subdomains. (TeXed notes, handout)
Cover [Bos] Sect. 3.3. Understand the canonical topology of affinoid spaces and their affinoid subdomains, in particular, prove Prop. 11. Moreover, discuss some further properties about affinoid subdomains: Prop 12-Thm 20.
(1/16) Xiaozheng Han: Affinoid functions I. (TeXed notes)
Cover [Bos] P65-P76. Describe the canonical presheaf and its stalks $\mathcal{O}_{X,x}$. Prove Sect 4.1 Prop. 2 and Prop 6. Then discuss locally closed immersion and Runge immersion. Finally, state extension lemma 9 for the preparation of next talk.
(1/19) Xingzhu Fang: Affinoid functions II. (Wenhan away, no notes for this talk)
Cover [Bos] P76-P91. Understand Gerritzen-Grauert theorem (Sect. 4.2, Thm. 10) and sketch the proof. Understand the Tate’s acyclic theorem (Sect. 4.3, Thm. 1) and sketch the proof. If time permits, discuss the generalized version of Tate’s acyclic theorem (Thm. 10, Cor. 11) (For more details, see [BGR] Chap. 8).
(1/30) Tianwei Gao: Grothendieck topologies and Rigid spaces. (TeXed notes, speaker’s handout, appendix)
Cover [Bos] P93-P106. Define Grothendieck topology and understand the strong Grothendieck topologies on affinoid spaces. In particular, prove Sect. 5.1, Cor. 10. Then discuss the sheaves on $G$-topological spaces and prove Sect. 5.2, Prop. 4. Finally, define the rigid $K$-spaces.
(2/2) Haoda Li: The GAGA functor. (TeXed notes, speaker’s handout)
[Bos] P106-P116. Finish Sect. 5.3. Construct the rigid analytifications of $K$-schemes of locally finite type and prove Sect. 5.4. Prop. 4. Then discuss the examples in the rest of Sect. 5.4.
(2/6) Yu Xiao: Coherent sheaves on rigid spaces. (speaker’s handout)
Cover [Bos] Sect. 5.1-5.3. Discuss the coherent sheaves of rigid spaces and sketch the proof of Sect. 5.1. Thm. 4. Then discuss the cohomologies of $\mathcal{O}_X$-modules. Finally, understand proper morphisms of rigid spaces and the proper mapping theorem.