Added Lecture 1 for Rigid Analytic Geometry

.github/workflows/pdf.yml

@@ -15,6 +15,7 @@ jobs:
       FILES: |
         F4D1-Analysis-and-Geometry-on-Manifolds/Analysis_on_Manifolds_Notes.tex
         V4A1-Algebraic-Geometry-I/Algebraic_Geometry_I_Notes.tex
+        V4A5-Rigid-Analytic-Geometry/Rigid_Analytic_Geometry_Notes.tex
         V5A2-Habiro-Rings/Habiro_Rings_Notes.tex
         V5B1-Advanced-Topics-in-Complex-Analysis/Advanced_Complex_Analysis_Notes.tex
     steps:

.gitignore

@@ -305,5 +305,6 @@ TSWLatexianTemp*
 #*Notes.bib
 F4D1-Analysis-and-Geometry-on-Manifolds/Analysis_on_Manifolds_Notes.pdf
 V4A1-Algebraic-Geometry-I/Algebraic_Geometry_I_Notes.pdf
+V4A5-Rigid-Analytic-Geometry/Rigid_Analytic_Geometry_Notes.pdf
 V5A2-Habiro-Rings/Habiro_Rings_Notes.pdf
-V5B1-Advanced-Topics-in-Complex-Analysis/Advanced_Complex_Analysis_Notes.pdf
+V5B1-Advanced-Topics-in-Complex-Analysis/Advanced_Complex_Analysis_Notes.pdf

F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 1.tex

@@ -103,7 +103,7 @@ \section{Lecture 1 -- 8th October 2024}\label{sec: lecture 1}
     Let $X$ be a topological space. If $X$ is second countable then any open cover of $X$ admits a countable subcover. 
 \end{proposition}
 \begin{proof}
-    Let $\Bcal$ be a countable basis for $X$ and $\{U_{i}\}_{i\in I}$ an open cover of $X$. Consider $\widetilde{\Bcal}$ consisting of those basis elements of $X$ contained in some $U_{i}$. Note that $\widetilde{\Bcal}$ is a cover of $X$ since for any point $x\in U_{i}$ there is an element of $\Bcal$ containing $x$ contained in $U_{i}$. For each $V\in\widetilde{\Bcal}$ of which there are countably many, consider $U_{V}\in\{U_{i}\}_{i\in I}$ such that $V\subseteq U_{V}$ these form a cover of $X$ indexed by a countable set $\widetilde{\Bcal}$ giving the claim. 
+    Let $\Bcal$ be a countable basis for $X$ and $\{U_{i}\}_{i\in I}$ an open cover of $X$. Consider $\widetilde{\Bcal}$ consisting of those basis elements of $X$ contained in some $U_{i}$. Note that $\widetilde{\Bcal}$ is a cover of $X$ since for any point $x\in U_{i}$ there is an element of $\Bcal$ containing $x$ contained in $U_{i}$. For each $V\in\widetilde{\Bcal}$ of which there are countably many, consider $U_{V}\in\{U_{i}\}_{i\in I}$ such that $V\subseteq U_{V}$. These form a cover of $X$ indexed by a countable set $\widetilde{\Bcal}$ giving the claim. 
 \end{proof}
 We also introduce the following notion of compact exhaustability. 
 \begin{definition}[Compact Exhaustability]\label{def: compact exhaustability}

V4A5-Rigid-Analytic-Geometry/Lecture 1.tex

@@ -0,0 +1,29 @@
+\section{Lecture 1 -- 11th October 2024}\label{sec: lecture 1}
+The goal of this course is to develop a theory of analytic functions in several variables over non-Archimedean fields -- analogous to the theory of functions in several complex variables -- compatible with the algebraic geometry over such fields in the style of Serre's GAGA principle. 
+
+Several problems arise from this desideratum:
+\begin{itemize}
+    \item The topology on $\QQ_{p}$ and $\CC_{p}$, the $p$-completion of its algebraic closure, are highly disconnected. 
+    \item It is not reasonable for the germs of an analytic function to determine the global behavior of the function. 
+    \item The unit disc $\{z\in\CC_{p}:|z|<1\}\subseteq\CC_{p}$ is non-compact. 
+\end{itemize}
+John Tate proposed a solution to this problem was to force certain sets to become quasicompact,\marginpar{Analogues of this theory are also visible in real algebraic geometry as developed by the school of Delfs-Knebusch \cite{DK}.} a perspective elaborated on by Bosch, Grauert, Gr\"{u}nzer, and Remmert (vis. \cite{BGR}). This was complemented by an alternative approach employed by Fresnel and van der Put which added points to the relevant spaces, later further developed by Berkovich and Huber (vis. \cite{Berkovich,FvdP,Huber}). 
+
+We will largely follow the approach of Tate and his followers, highlighting contemporary applications. 
+
+We now make the following recollections from point set topology. We will largely ignore set-theoretic issues throughout. 
+\begin{definition}[Sieve]\label{def: sieve}
+    Let $X$ be an object of a category $\Csf$. A sieve on $X$ is a collection of morphisms with target $X$ that is closed under composition on the left: if $f:Y\to X$ is in the sieve, then $g\circ f:Y'\to X$ is in the sieve for all $g:Y'\to Y$. 
+\end{definition}
+Here are some examples.
+\begin{definition}[All Sieve]\label{def: all sieve}
+    Let $X$ be an object of a category $\Csf$. The all sieve on $X$ consists of all morphisms in $\Csf$ with target $X$. 
+\end{definition}
+\begin{definition}[Empty Sieve]\label{def: empty sieve}
+    Let $X$ be an object of a category $\Csf$. The empty sieve on $X$ consists of no morphisms. 
+\end{definition}
+Given a collection of morphisms with fixed target, we can construct a sieve as follows. 
+\begin{definition}[Sieve Generated by a Collection]\label{def: sieve generated by a collection}
+    Let $X$ be an object of a category $\Csf$ and $\{f_{i}:X_{i}\to X\}_{i\in I}$ a collection of morphisms in $\Csf$ with target $X$. The sieve generated by this collection consists of those morphisms $g:Y\to X$ such that $g$ factors through some $X_{i}$.\marginpar{Explicitly, there exists some $i\in I$ and a map $\gamma:Y\to X_{i}$ such that $g=f_{i}\circ\gamma$.}
+\end{definition}
+We should think of sieves as taking away objects that are ``small enough'' with respect to $X$ in the sense that they admit a map to $X$.  

V4A5-Rigid-Analytic-Geometry/Rigid_Analytic_Geometry_Notes.tex

@@ -0,0 +1,51 @@
+\documentclass{amsart}
+\usepackage[margin=1.5in]{geometry} 
+\usepackage{amsmath}
+\usepackage{tcolorbox}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{lastpage}
+\usepackage{fancyhdr}
+\usepackage{accents}
+\usepackage{hyperref}
+\usepackage{xcolor}
+\usepackage{color}
+\input{shortcuts.tex}
+\setlength{\headheight}{40pt}
+
+
+\newenvironment{solution}
+  {\renewcommand\qedsymbol{$\blacksquare$}
+  \begin{proof}[Solution]}
+  {\end{proof}}
+\renewcommand\qedsymbol{$\blacksquare$}
+
+\usepackage{amsmath, amssymb, tikz, amsthm, csquotes, multicol, footnote, tablefootnote, biblatex, wrapfig, float, quiver, mathrsfs, cleveref, enumitem, upgreek, stmaryrd, marginnote}
+\addbibresource{refs.bib}
+\theoremstyle{definition}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{exercise}[theorem]{Exercise}
+\newtheorem{question}[theorem]{Question}
+\newtheorem{example}[theorem]{Example}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{conjecture}[theorem]{Conjecture}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{definition}[theorem]{Definition}
+\numberwithin{equation}{section}
+\begin{document}
+\large
+\title[Rigid Analytic Geometry]{V4A5 -- Rigid Analytic Geometry \\ Winter Semester 2024/25}
+\author{Wern Juin Gabriel Ong}
+\address{Universit\"{a}t Bonn, Bonn, D-53111}
+\email{[email protected]}
+\urladdr{https://wgabrielong.github.io/}
+\maketitle
+\section*{Preliminaries}
+These notes roughly correspond to the course \textbf{V4A5 -- Rigid Analytic Geometry} taught by Prof. Jens Franke at the Universit\"{a}t Bonn in the Winter 2024/25 semester. These notes are \LaTeX-ed after the fact with significant alteration and are subject to misinterpretation and mistranscription. Use with caution. Any errors are undoubtedly my own and any virtues that could be ascribed to these notes ought be attributed to the instructor and not the typist. 
+\tableofcontents
+\include{Lecture 1}
+\newpage
+\printbibliography
+\end{document}

V4A5-Rigid-Analytic-Geometry/quiver.sty

@@ -0,0 +1,40 @@
+% *** quiver ***
+% A package for drawing commutative diagrams exported from https://q.uiver.app.
+%
+% This package is currently a wrapper around the `tikz-cd` package, importing necessary TikZ
+% libraries, and defining a new TikZ style for curves of a fixed height.
+%
+% Version: 1.2.2
+% Authors:
+% - varkor (https://github.com/varkor)
+% - AndréC (https://tex.stackexchange.com/users/138900/andr%C3%A9c)
+
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{quiver}[2021/01/11 quiver]
+
+% `tikz-cd` is necessary to draw commutative diagrams.
+\RequirePackage{tikz-cd}
+% `amssymb` is necessary for `\lrcorner` and `\ulcorner`.
+\RequirePackage{amssymb}
+% `calc` is necessary to draw curved arrows.
+\usetikzlibrary{calc}
+% `pathmorphing` is necessary to draw squiggly arrows.
+\usetikzlibrary{decorations.pathmorphing}
+
+% A TikZ style for curved arrows of a fixed height, due to AndréC.
+\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
+    .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
+    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
+    .. (\tikztotarget)\tikztonodes}},
+    settings/.code={\tikzset{quiver/.cd,#1}
+        \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
+    quiver/.cd,pos/.initial=0.35,height/.initial=0}
+
+% TikZ arrowhead/tail styles.
+\tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}}
+\tikzset{2tail/.code={\pgfsetarrowsstart{Implies[reversed]}}}
+\tikzset{2tail reversed/.code={\pgfsetarrowsstart{Implies}}}
+% TikZ arrow styles.
+\tikzset{no body/.style={/tikz/dash pattern=on 0 off 1mm}}
+
+\endinput

V4A5-Rigid-Analytic-Geometry/refs.bib

@@ -0,0 +1,83 @@
+@Book{BGR,
+ Author = {Bosch, Siegfried and G{\"u}ntzer, Ulrich and Remmert, Reinhold},
+ Title = {Non-{Archimedean} analysis. {A} systematic approach to rigid analytic geometry},
+ FSeries = {Grundlehren der Mathematischen Wissenschaften},
+ Series = {Grundlehren Math. Wiss.},
+ ISSN = {0072-7830},
+ Volume = {261},
+ Year = {1984},
+ Publisher = {Springer, Cham},
+ Language = {English},
+ Keywords = {14G20,32P05,12J10,32-02,14-02,32B05,46S10},
+ zbMATH = {3857279},
+ Zbl = {0539.14017},
+ shorthand = {BGR84}
+}
+
+@Book{FvdP,
+ Author = {Fresnel, Jean and van der Put, Marius},
+ Title = {Rigid analytic geometry and its applications},
+ FSeries = {Progress in Mathematics},
+ Series = {Prog. Math.},
+ ISSN = {0743-1643},
+ Volume = {218},
+ ISBN = {0-8176-4206-4},
+ Year = {2004},
+ Publisher = {Boston, MA: Birkh{\"a}user},
+ Language = {English},
+ Keywords = {14G22,30H05,32P05,30G06},
+ zbMATH = {2043955},
+ Zbl = {1096.14014},
+ shorthand = {FvdP04}
+}
+
+@Book{Berkovich,
+ Author = {Berkovich, Vladimir G.},
+ Title = {Spectral theory and analytic geometry over non-{Archimedean} fields},
+ FSeries = {Mathematical Surveys and Monographs},
+ Series = {Math. Surv. Monogr.},
+ ISSN = {0076-5376},
+ Volume = {33},
+ ISBN = {0-8218-1534-2},
+ Year = {1990},
+ Publisher = {Providence, RI: American Mathematical Society},
+ Language = {English},
+ Keywords = {14G20,32P05},
+ zbMATH = {47872},
+ Zbl = {0715.14013},
+ shorthand = {Ber90}
+}
+
+@Book{Huber,
+ Author = {Huber, Roland},
+ Title = {{\'E}tale cohomology of rigid analytic varieties and adic spaces},
+ FSeries = {Aspects of Mathematics},
+ Series = {Aspects Math.},
+ ISSN = {0179-2156},
+ Volume = {E30},
+ ISBN = {3-528-06794-2},
+ Year = {1996},
+ Publisher = {Wiesbaden: Vieweg},
+ Language = {English},
+ Keywords = {14F20,32P05},
+ zbMATH = {981434},
+ Zbl = {0868.14010},
+ shorthand = {Hub96}
+}
+
+@Book{DK,
+ Author = {Delfs, Hans and Knebusch, Manfred},
+ Title = {Locally semialgebraic spaces},
+ FSeries = {Lecture Notes in Mathematics},
+ Series = {Lect. Notes Math.},
+ ISSN = {0075-8434},
+ Volume = {1173},
+ Year = {1985},
+ Publisher = {Springer, Cham},
+ Language = {English},
+ DOI = {10.1007/BFb0074551},
+ Keywords = {14Pxx,14-02,14F45},
+ zbMATH = {3931165},
+ Zbl = {0582.14006},
+ shorthand = {DK85}
+}

V4A5-Rigid-Analytic-Geometry/shortcuts.tex

@@ -0,0 +1,104 @@
+% Fields
+\newcommand{\CC}{\mathbb{C}}
+\newcommand{\RR}{\mathbb{R}}
+\newcommand{\QQ}{\mathbb{Q}}
+\newcommand{\ZZ}{\mathbb{Z}}
+\newcommand{\NN}{\mathbb{N}}
+\newcommand{\FF}{\mathbb{F}}
+\newcommand{\PP}{\mathbb{P}}
+
+% mathcal letters
+\newcommand{\Acal}{\mathcal{A}}
+\newcommand{\Bcal}{\mathcal{B}}
+\newcommand{\Ccal}{\mathcal{C}}
+\newcommand{\Dcal}{\mathcal{D}}
+\newcommand{\Ecal}{\mathcal{E}}
+\newcommand{\Fcal}{\mathcal{F}}
+\newcommand{\Gcal}{\mathcal{G}}
+\newcommand{\Hcal}{\mathcal{H}}
+\newcommand{\Ical}{\mathcal{I}}
+\newcommand{\Jcal}{\mathcal{J}}
+\newcommand{\Kcal}{\mathcal{K}}
+\newcommand{\Lcal}{\mathcal{L}}
+\newcommand{\Mcal}{\mathcal{M}}
+\newcommand{\Ncal}{\mathcal{N}}
+\newcommand{\Ocal}{\mathcal{O}}
+\newcommand{\Pcal}{\mathcal{P}}
+\newcommand{\Qcal}{\mathcal{Q}}
+\newcommand{\Rcal}{\mathcal{R}}
+\newcommand{\Scal}{\mathcal{S}}
+\newcommand{\Tcal}{\mathcal{T}}
+\newcommand{\Ucal}{\mathcal{U}}
+\newcommand{\Vcal}{\mathcal{V}}
+\newcommand{\Wcal}{\mathcal{W}}
+\newcommand{\Xcal}{\mathcal{X}}
+\newcommand{\Ycal}{\mathcal{Y}}
+\newcommand{\Zcal}{\mathcal{Z}}
+
+% abstract categories
+\newcommand{\Asf}{\mathsf{A}}
+\newcommand{\Bsf}{\mathsf{B}}
+\newcommand{\Csf}{\mathsf{C}}
+\newcommand{\Dsf}{\mathsf{D}}
+\newcommand{\Esf}{\mathsf{E}}
+\newcommand{\Ssf}{\mathsf{S}}
+
+% algebraic geometry
+\newcommand{\spec}{\operatorname{Spec}}
+\newcommand{\proj}{\operatorname{Proj}}
+\newcommand{\res}{\mathrm{res}}
+
+% categories 
+\newcommand{\id}{\mathrm{id}}
+\newcommand{\Obj}{\mathrm{Obj}}
+\newcommand{\Mor}{\mathrm{Mor}}
+\newcommand{\Hom}{\mathrm{Hom}}
+\newcommand{\Aut}{\mathrm{Aut}}
+\newcommand{\Sets}{\mathsf{Sets}}
+\newcommand{\SSets}{\mathsf{SSets}}
+\newcommand{\kVect}{\mathsf{Vect}_{k}}
+\newcommand{\Vect}{\mathsf{Vect}}
+\newcommand{\Alg}{\mathsf{Alg}}
+\newcommand{\Ring}{\mathsf{Ring}}
+\newcommand{\Mod}{\mathsf{Mod}}
+\newcommand{\Grp}{\mathsf{Grp}}
+\newcommand{\AbGrp}{\mathsf{AbGrp}}
+\newcommand{\PSh}{\mathsf{PSh}}
+\newcommand{\Sh}{\mathsf{Sh}}
+\newcommand{\PSch}{\mathsf{PSch}}
+\newcommand{\Sch}{\mathsf{Sch}}
+\newcommand{\Top}{\mathsf{Top}}
+\newcommand{\Com}{\mathsf{Com}}
+\newcommand{\Coh}{\mathsf{Coh}}
+\newcommand{\QCoh}{\mathsf{QCoh}}
+\newcommand{\Opens}{\mathsf{Opens}}
+\newcommand{\Opp}{\mathsf{Opp}}
+\newcommand{\Cat}{\mathsf{Cat}}
+\newcommand{\NatTrans}{\mathrm{NatTrans}}
+\newcommand{\pr}{\mathrm{pr}}
+\newcommand{\Fun}{\mathrm{Fun}}
+\newcommand{\colim}{\mathrm{colim}}
+\newcommand{\lifts}{\boxslash}
+\DeclareMathOperator\squarediv{\lifts}
+\newcommand{\Kan}{\mathsf{Kan}}
+\newcommand{\Path}{\mathsf{Path}}
+\newcommand{\SPSh}{\mathsf{SPSh}}
+\newcommand{\SSh}{\mathsf{SSh}}
+\newcommand{\Bord}{\mathsf{Bord}}
+
+% simplicial sets
+\newcommand{\DDelta}{\Updelta}
+\newcommand{\Sing}{\operatorname{Sing}}
+
+% ideal theory
+\newcommand{\mfrak}{\mathfrak{m}}
+\newcommand{\afrak}{\mathfrak{a}}
+\newcommand{\bfrak}{\mathfrak{b}}
+\newcommand{\pfrak}{\mathfrak{p}}
+\newcommand{\qfrak}{\mathfrak{q}}
+
+% number theory
+\newcommand{\Tr}{\mathrm{Tr}}
+\newcommand{\Nm}{\mathrm{Nm}}
+\newcommand{\Gal}{\mathrm{Gal}}
+\newcommand{\Frob}{\mathrm{Frob}}