Compare commits
31
Commits
f8ee1c4a1c
...
master
| Author | SHA1 | Date | |
|---|---|---|---|
 christian
|
104be68bf7
|
||
|
christian
|
b90d3dd001
|
||
|
christian
|
fe2412987f
|
||
|
christian
|
0a85566e56
|
||
|
christian
|
c2628bde37
|
||
|
christian
|
90b01a491f
|
||
|
christian
|
52a55a6c6f
|
||
|
christian
|
4c32de62c4
|
||
|
christian
|
b9afb52c4b
|
||
|
christian
|
6bdb35767e
|
||
|
christian
|
1544563802
|
||
|
christian
|
f929db6df0
|
||
|
christian
|
94e329d376
|
||
|
christian
|
2b4c8ec637
|
||
|
christian
|
f570713948
|
||
|
christian
|
25f7eb4ea3
|
||
|
christian
|
9ed7216425
|
||
|
christian
|
f5957abd5b
|
||
|
christian
|
59b1b8ce78
|
||
|
christian
|
8810bdeb33
|
||
|
christian
|
98e48b8d88
|
||
|
christian
|
09cab8543e
|
||
|
christian
|
f6e36c24b0
|
||
|
christian
|
f5a077e53f
|
||
|
christian
|
ba6b781a24
|
||
|
christian
|
e8b33bfa2f | ||
|
christian
|
01573c3de7 | ||
|
christian
|
e79ca0fe16 | ||
|
christian
|
92c0ac3814 | ||
|
christian
|
3b712b98ec | ||
|
christian
|
3b31abde5a |
@@ -17,3 +17,5 @@
|
||||
*.snm
|
||||
*.bbl
|
||||
*.blg
|
||||
*.table
|
||||
*.gnuplot
|
||||
|
||||
@@ -10,3 +10,12 @@
|
||||
[submodule "ws2020/wtheo/uebungen"]
|
||||
path = ws2020/wtheo/uebungen
|
||||
url = https://git.flavigny.de/christian/wtheo-zettel
|
||||
[submodule "ws2022/rav/lecture"]
|
||||
path = ws2022/rav/lecture
|
||||
url = git@git.mathi.uni-heidelberg.de:cmerten/ravlecture
|
||||
[submodule "ws2023/groupschemes"]
|
||||
path = ws2023/groupschemes
|
||||
url = https://git.flavigny.de/christian/groupschemes-lecture
|
||||
[submodule "ws2023/groupschemes-lecture"]
|
||||
path = ws2023/groupschemes-lecture
|
||||
url = https://git.flavigny.de/christian/groupschemes-lecture
|
||||
|
||||
@@ -0,0 +1,249 @@
|
||||
\ProvidesClass{notes}
|
||||
\LoadClass[a4paper]{amsart}
|
||||
|
||||
\RequirePackage[utf8]{inputenc}
|
||||
\RequirePackage[T1]{fontenc}
|
||||
\RequirePackage{textcomp}
|
||||
\RequirePackage[german, english]{babel}
|
||||
\RequirePackage{amsmath, amssymb, amsthm}
|
||||
\RequirePackage{mdframed}
|
||||
\RequirePackage{tikz-cd}
|
||||
\RequirePackage{fancyhdr}
|
||||
\RequirePackage{geometry}
|
||||
\RequirePackage{import}
|
||||
\RequirePackage{pdfpages}
|
||||
%\RequirePackage{transparent}
|
||||
\RequirePackage{xcolor}
|
||||
\RequirePackage{array}
|
||||
\RequirePackage[shortlabels]{enumitem}
|
||||
\RequirePackage{tikz}
|
||||
\RequirePackage{pgfplots}
|
||||
\RequirePackage{listings}
|
||||
\RequirePackage{mathtools}
|
||||
\RequirePackage{forloop}
|
||||
\RequirePackage{totcount}
|
||||
\RequirePackage{calc}
|
||||
\RequirePackage{wasysym}
|
||||
\RequirePackage{environ}
|
||||
\RequirePackage{hyperref}
|
||||
\RequirePackage{graphicx}
|
||||
|
||||
\usetikzlibrary{quotes, angles}
|
||||
\pgfplotsset{
|
||||
compat=1.15,
|
||||
default 2d plot/.style={%
|
||||
grid=both,
|
||||
minor tick num=4,
|
||||
grid style={line width=.1pt, draw=gray!10},
|
||||
major grid style={line width=.2pt,draw=gray!50},
|
||||
axis lines=middle,
|
||||
enlargelimits={abs=0.2}
|
||||
},
|
||||
}
|
||||
|
||||
\usetikzlibrary{quotes, angles, math}
|
||||
\pgfplotsset{
|
||||
compat=1.15,
|
||||
axis lines = middle,
|
||||
ticks = none,
|
||||
%default 2d plot/.style={%
|
||||
% ticks=none,
|
||||
% axis lines = middle,
|
||||
% grid=both,
|
||||
% minor tick num=4,
|
||||
% grid style={line width=.1pt, draw=gray!10},
|
||||
% major grid style={line width=.2pt,draw=gray!50},
|
||||
% axis lines=middle,
|
||||
% enlargelimits={abs=0.2}
|
||||
}
|
||||
|
||||
\newcounter{curve}
|
||||
|
||||
\NewDocumentCommand{\algebraiccurve}{ O{} O{$#5 = 0$} O{-4:4} O{-4:4} m }{
|
||||
\addplot[id=curve\arabic{curve}, raw gnuplot, smooth, #1] function{%
|
||||
f(x,y) = #5;
|
||||
set xrange [#3];
|
||||
set yrange [#4];
|
||||
set view 0,0;
|
||||
set isosample 1000,1000;
|
||||
set size square;
|
||||
set cont base;
|
||||
set cntrparam levels incre 0,0.1,0;
|
||||
unset surface;
|
||||
splot f(x,y)
|
||||
};
|
||||
\addlegendentry{#2}
|
||||
\stepcounter{curve}
|
||||
}%
|
||||
|
||||
% PAGE GEOMETRY
|
||||
\geometry{
|
||||
top=1.2in,bottom=1.4in,left=1.3in,right=1.3in,
|
||||
bottom=35mm
|
||||
}
|
||||
|
||||
% PARAGRAPH no indent but skip
|
||||
%\setlength{\parskip}{3mm}
|
||||
%\setlength{\parindent}{0mm}
|
||||
|
||||
\newtheorem{satz}{Proposition}[section]
|
||||
\newtheorem{theorem}[satz]{Theorem}
|
||||
\newtheorem{lemma}[satz]{Lemma}
|
||||
\newtheorem{korollar}[satz]{Corollary}
|
||||
\theoremstyle{definition}
|
||||
\newtheorem{definition}[satz]{Definition}
|
||||
\newtheorem*{definition*}{Definition}
|
||||
|
||||
%\theoremstyle{definition}
|
||||
%\newmdtheoremenv{satz}{Satz}[section]
|
||||
%\newmdtheoremenv{lemma}[satz]{Lemma}
|
||||
%\newmdtheoremenv{korollar}[satz]{Korollar}
|
||||
%\newmdtheoremenv{definition}[satz]{Definition}
|
||||
|
||||
\newtheorem{bsp}[satz]{Example}
|
||||
\newtheorem{bem}[satz]{Remark}
|
||||
\newtheorem{aufgabe}{Exercise}
|
||||
|
||||
% enable aufgaben counting
|
||||
\regtotcounter{aufgabe}
|
||||
|
||||
% temporary calculation counter
|
||||
\newcounter{var}
|
||||
|
||||
\newcommand{\N}{\mathbb{N}}
|
||||
\newcommand{\R}{\mathbb{R}}
|
||||
\newcommand{\Z}{\mathbb{Z}}
|
||||
\newcommand{\Q}{\mathbb{Q}}
|
||||
\newcommand{\C}{\mathbb{C}}
|
||||
|
||||
% HEADERS
|
||||
|
||||
\pagestyle{headings}
|
||||
|
||||
\newcommand{\incfig}[1]{%
|
||||
\def\svgwidth{\columnwidth}
|
||||
\import{./figures/}{#1.pdf_tex}
|
||||
}
|
||||
\pdfsuppresswarningpagegroup=1
|
||||
|
||||
% code listings, define style
|
||||
\lstdefinestyle{mystyle}{
|
||||
commentstyle=\color{gray},
|
||||
keywordstyle=\color{blue},
|
||||
numberstyle=\tiny\color{gray},
|
||||
stringstyle=\color{black},
|
||||
basicstyle=\ttfamily\footnotesize,
|
||||
breakatwhitespace=false,
|
||||
breaklines=true,
|
||||
captionpos=b,
|
||||
keepspaces=true,
|
||||
numbers=left,
|
||||
numbersep=5pt,
|
||||
showspaces=false,
|
||||
showstringspaces=false,
|
||||
showtabs=false,
|
||||
tabsize=2
|
||||
}
|
||||
|
||||
% activate my colour style
|
||||
\lstset{style=mystyle}
|
||||
|
||||
% better stackrel
|
||||
\let\oldstackrel\stackrel
|
||||
\renewcommand{\stackrel}[2]{%
|
||||
\oldstackrel{\mathclap{#1}}{#2}
|
||||
}%
|
||||
|
||||
% integral d sign
|
||||
\makeatletter \renewcommand\d[2][]{\ensuremath{%
|
||||
\,\mathrm{d}^{#1}#2\@ifnextchar^{}{\@ifnextchar\d{}{\,}}}}
|
||||
\makeatother
|
||||
|
||||
% contradiction
|
||||
\newcommand{\contr}{\text{\Large\lightning}}
|
||||
|
||||
% disjoint unions: provides cupdot and bigcupdot
|
||||
\makeatletter
|
||||
\def\moverlay{\mathpalette\mov@rlay}
|
||||
\def\mov@rlay#1#2{\leavevmode\vtop{%
|
||||
\baselineskip\z@skip \lineskiplimit-\maxdimen
|
||||
\ialign{\hfil$\m@th#1##$\hfil\cr#2\crcr}}}
|
||||
\newcommand{\charfusion}[3][\mathord]{
|
||||
#1{\ifx#1\mathop\vphantom{#2}\fi
|
||||
\mathpalette\mov@rlay{#2\cr#3}
|
||||
}
|
||||
\ifx#1\mathop\expandafter\displaylimits\fi}
|
||||
\makeatother
|
||||
|
||||
\newcommand{\cupdot}{\charfusion[\mathbin]{\cup}{\cdot}}
|
||||
\newcommand{\bigcupdot}{\charfusion[\mathop]{\bigcup}{\cdot}}
|
||||
|
||||
\ExplSyntaxOn
|
||||
|
||||
% S-tackrelcompatible ALIGN environment
|
||||
% some might also call it the S-uper ALIGN environment
|
||||
% uses regular expressions to calculate the widest stackrel
|
||||
% to put additional padding on both sides of relation symbols
|
||||
\NewEnviron{salign}
|
||||
{
|
||||
\begin{align}
|
||||
\lec_insert_padding:V \BODY
|
||||
\end{align}
|
||||
}
|
||||
% starred version that does no equation numbering
|
||||
\NewEnviron{salign*}
|
||||
{
|
||||
\begin{align*}
|
||||
\lec_insert_padding:V \BODY
|
||||
\end{align*}
|
||||
}
|
||||
|
||||
% some helper variables
|
||||
\tl_new:N \l__lec_text_tl
|
||||
\seq_new:N \l_lec_stackrels_seq
|
||||
\int_new:N \l_stackrel_count_int
|
||||
\int_new:N \l_idx_int
|
||||
\box_new:N \l_tmp_box
|
||||
\dim_new:N \l_tmp_dim_a
|
||||
\dim_new:N \l_tmp_dim_b
|
||||
\dim_new:N \l_tmp_dim_needed
|
||||
|
||||
% function to insert padding according to widest stackrel
|
||||
\cs_new_protected:Nn \lec_insert_padding:n
|
||||
{
|
||||
\tl_set:Nn \l__lec_text_tl { #1 }
|
||||
% get all stackrels in this align environment
|
||||
\regex_extract_all:nnN { \c{stackrel}{(.*?)}{(.*?)} } { #1 } \l_lec_stackrels_seq
|
||||
% get number of stackrels
|
||||
\int_set:Nn \l_stackrel_count_int { \seq_count:N \l_lec_stackrels_seq }
|
||||
\int_set:Nn \l_idx_int { 1 }
|
||||
\dim_set:Nn \l_tmp_dim_needed { 0pt }
|
||||
% iterate over stackrels
|
||||
\int_while_do:nn { \l_idx_int <= \l_stackrel_count_int }
|
||||
{
|
||||
% calculate width of text
|
||||
\hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 1 }$}
|
||||
\dim_set:Nn \l_tmp_dim_a {\box_wd:N \l_tmp_box}
|
||||
% calculate width of relation symbol
|
||||
\hbox_set:Nn \l_tmp_box {$\seq_item:Nn \l_lec_stackrels_seq { \l_idx_int + 2 }$}
|
||||
\dim_set:Nn \l_tmp_dim_b {\box_wd:N \l_tmp_box}
|
||||
% check if 0.5*(a-b) > minimum padding, if yes updated minimum padding
|
||||
\dim_compare:nNnTF
|
||||
{ 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } > { \l_tmp_dim_needed }
|
||||
{ \dim_set:Nn \l_tmp_dim_needed { 1pt * \dim_ratio:nn { \l_tmp_dim_a - \l_tmp_dim_b } { 2pt } } }
|
||||
{ }
|
||||
\quad
|
||||
% increment list index by three, as every stackrel produces three list entries
|
||||
\int_incr:N \l_idx_int
|
||||
\int_incr:N \l_idx_int
|
||||
\int_incr:N \l_idx_int
|
||||
}
|
||||
% replace all relations with align characters (&) and add the needed padding
|
||||
\regex_replace_all:nnN
|
||||
{ (<&|&<|\c{iff}&|&\c{iff}|\c{impliedby}&|&\c{impliedby}|\c{implies}&|&\c{implies}|\c{approx}&|&\c{approx}|\c{equiv}&|&\c{equiv}|=&|&=|\c{le}&|&\c{le}|\c{ge}&|&\c{ge}|&\c{stackrel}{.*?}{.*?}|\c{stackrel}{.*?}{.*?}&|&\c{neq}|\c{neq}&|\c{simeq}&|&\c{simeq}) }
|
||||
{ \c{kern} \u{l_tmp_dim_needed} \1 \c{kern} \u{l_tmp_dim_needed} }
|
||||
\l__lec_text_tl
|
||||
\l__lec_text_tl
|
||||
}
|
||||
\cs_generate_variant:Nn \lec_insert_padding:n { V }
|
||||
\ExplSyntaxOff
|
||||
Binary file not shown.
@@ -0,0 +1,169 @@
|
||||
\documentclass[a4paper, 11pt]{article}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage{textcomp}
|
||||
\usepackage[german]{babel}
|
||||
\usepackage{amsmath, amssymb, amsthm}
|
||||
\usepackage[a4 paper,margin=1in]{geometry}
|
||||
\usepackage{fancyhdr}
|
||||
\usepackage{tikz-cd}
|
||||
\usepackage{enumerate}
|
||||
\pagestyle{fancy}
|
||||
\fancyhead{}
|
||||
\fancyfoot{}
|
||||
\rhead{30.6.2022}
|
||||
\makeatletter
|
||||
\newcommand{\colim@}[2]{%
|
||||
\vtop{\m@th\ialign{##\cr
|
||||
\hfil$#1\operator@font colim$\hfil\cr
|
||||
\noalign{\nointerlineskip\kern1.5\ex@}#2\cr
|
||||
\noalign{\nointerlineskip\kern-\ex@}\cr}}%
|
||||
}
|
||||
\newcommand{\colim}{%
|
||||
\mathop{\mathpalette\colim@{\rightarrowfill@\textstyle}}\nmlimits@
|
||||
}
|
||||
\makeatother
|
||||
|
||||
\newcommand{\spec}{\operatorname{Spec }}
|
||||
\newtheorem{satz}{Satz}
|
||||
\newtheorem{thm}[satz]{Theorem}
|
||||
\newtheorem{lemma}[satz]{Lemma}
|
||||
\newtheorem{definition}[satz]{Definition}
|
||||
\newtheorem{bem}[satz]{Bemerkung}
|
||||
\newtheorem{bung}[satz]{Übung}
|
||||
\newtheorem{rem}[satz]{Erinnerung}
|
||||
\begin{document}
|
||||
\section*{Endlich étale Morphismen, Vortrag 9}
|
||||
|
||||
Sei $A$ ein (kommutativer) Ring.
|
||||
% \begin{rem}[Komposition]
|
||||
% Sei $B$ endliche, projektive $A$-Algebra und $C$ endliche, projektive $B$-Algebra. Dann
|
||||
% ist $C$ endliche, projektive $A$-Algebra.
|
||||
% \label{satz:composition-projective}
|
||||
%\end{rem}
|
||||
|
||||
\begin{rem}%[Basiswechsel endlich projektive]
|
||||
Endlich (treu-)projektiv ist stabil unter Basiswechsel und Komposition. Insbesondere
|
||||
kommutiert für $A \to B$ endlich projektiv und $A \to C$ Ringhomomorphismus das folgende Diagramm:
|
||||
%Sei $B$ endlich projektive $A$-Algebra und $C$ eine weitere $A$-Algebra. Dann
|
||||
%ist $B \otimes_A C$ endlich projektive $C$-Algebra.
|
||||
%Insbesondere kommutiert das folgende Diagramm
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\spec C \arrow[swap]{dr}{[B \otimes_A C : C]} \arrow[from=1-1,to=1-3] & & \spec A \arrow{dl}{[B : A]} \\
|
||||
& \mathbb{Z} &.
|
||||
\end{tikzcd}
|
||||
\]
|
||||
\label{satz:basischange-projective}
|
||||
\end{rem}
|
||||
\begin{bung}
|
||||
Sei $B$ endliche, projektive $A$-Algebra. Dann ist
|
||||
\begin{enumerate}[(a)]
|
||||
\item $B = 0 \iff [B : A] = 0$.
|
||||
\item $A \to B$ Isomorphismus $\iff [B : A] = 1$.
|
||||
\item $\spec B \to \spec A$ surjektiv $\iff$ $B$ treuprojektive $A$-Algebra.
|
||||
\end{enumerate}
|
||||
\label{satz:degree}
|
||||
\end{bung}
|
||||
\begin{definition}[Zariskiüberdeckung]
|
||||
Wir nennen Elemente $\{f_i\}_{i \in I} \subseteq A$ eine Zariskiüberdeckung von $A$, wenn
|
||||
$\spec A = \bigcup_{i \in I} D(f_i)$.
|
||||
\end{definition}
|
||||
|
||||
% \begin{definition}
|
||||
% Sei $f\colon A \to B$ ein Ringhomomorphismus.
|
||||
% $f$ ist \emph{endlich und lokal frei}, wenn eine Zariskiüberdeckung existiert, sodass
|
||||
% $B_{f_i}$ endliche, freie $A_{f_{i}}$ Algebra ist für alle $i \in I$.
|
||||
% \label{def:finite-locally-free}
|
||||
%\end{definition}
|
||||
\begin{satz}[Äquivalente Charakterisierungen von endlich étale]
|
||||
Sei $A \to B$ ein Ringhomomorphismus. Dann sind äquivalent
|
||||
\begin{enumerate}[(i)]
|
||||
\item $A \to B$ endlich étale, das heißt endlich, flach und unverzweigt.
|
||||
%\item $B$ ist endlich präsentiert und flach als $A$-Modul und $\Omega_{B / A} = 0$.
|
||||
%\item $B$ ist endliche, projektive und separable $A$-Algebra.
|
||||
\item $A \to B$ endlich, projektiv und separabel.
|
||||
%\item $A \to B$ endlich, projektiv und $\Omega_{B / A} = 0$.
|
||||
\item Es existiert eine Zariskiüberdeckung $\{f_i\}_{i \in I}$ von $A$, sodass
|
||||
$A_{f_i} \to B_{f_i}$ endlich étale ist für alle $i \in I$.
|
||||
\item Es existiert eine Zariskiüberdeckung $\{f_i\}_{i \in I}$ von $A$, sodass
|
||||
$A_{f_i} \to B_{f_i}$ endlich frei und separabel ist für alle $i \in I$.
|
||||
%\item Es existiert eine treuprojektive $A$-Algebra $C$, sodass $B \otimes_A C$ total zerlegbar ist.
|
||||
\end{enumerate}
|
||||
\end{satz}
|
||||
|
||||
\begin{satz}%[Basiswechsel endlich étale]
|
||||
%Sei $B$ endlich étale $A$-Algebra und $C$ eine weitere $A$-Algebra. Dann ist
|
||||
%$B \otimes_{A} C$ endlich étale $C$-Algebra.
|
||||
%\label{satz:basischange}
|
||||
Endlich étale ist stabil unter Basiswechsel und Komposition.
|
||||
\end{satz}
|
||||
% \begin{satz}[Komposition endlich étale]
|
||||
% Sei $B$ endlich étale $A$-Algebra und $C$ endlich étale $B$-Algebra. Dann ist
|
||||
% $C$ endlich étale $A$-Algebra.
|
||||
%\end{satz}
|
||||
\begin{bung}
|
||||
Sei $A$ ein Ring und $(B_i)_{i \in I}$ $A$-Algebren mit $I$ endlich. Sei weiter
|
||||
$B = \prod_{i \in I} B_i$. Dann ist $B$ genau dann endlich étale, wenn
|
||||
jedes $B_i$ endlich étale ist. In diesem Fall gilt
|
||||
$[B : A] = \sum_{i \in I} [B_i : A]$.
|
||||
\label{ex:5.3}
|
||||
\end{bung}
|
||||
\begin{bung}
|
||||
Seien $(A_i)_{i \in I}, (B_i)_{i \in I}$ Ringe mit $I$ endlich und sei $B_i$ endlich étale $A_i$-Algebra
|
||||
für alle $i \in I$. Dann ist $\prod_{i \in I} B_i$ endlich étale $\prod_{i \in I} A_i$-Algebra. Außerdem
|
||||
ist jede endlich étale $\prod_{i \in I} A_i$ Algebra von dieser Form.
|
||||
Weiter ist
|
||||
\[
|
||||
\left[ \prod_{i \in I} B_i : \prod_{i \in I} A_i \right]\Big|_{\spec A_j} = [ B_j : A_j]
|
||||
.\]
|
||||
\label{satz:projective-prod}
|
||||
\end{bung}
|
||||
\begin{definition}[Total zerlegbare Algebren]
|
||||
Eine $A$-Algebra $B$ ist \emph{total zerlegbar}, wenn
|
||||
$A$ ein endliches Produkt von Ringen $A_n$ ist mit $n \ge 0$ und
|
||||
$B$ isomorph ist zu $\prod_{n \ge 0} A_n^{n}$, sodass
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
B \arrow{r}{\sim} & \prod_{n \ge 0}^{} A_n^{n} \\
|
||||
A \arrow{u} \arrow{r}{\sim} & \prod_{n \ge 0} A_n \arrow{u}
|
||||
\end{tikzcd}
|
||||
\] kommutiert.
|
||||
\end{definition}
|
||||
|
||||
\begin{thm}
|
||||
Sei $B$ eine $A$-Algebra. Dann ist $A \to B$ genau dann endlich étale,
|
||||
wenn eine treuprojektive $A$-Algebra $C$ existiert, sodass
|
||||
$C \to B \otimes_A C$ total zerlegbar ist.
|
||||
\label{th:5.10}
|
||||
\end{thm}
|
||||
|
||||
\begin{definition}
|
||||
Sei $E$ eine endliche Menge. Dann sei $A^{E} = \prod_{e \in E}^{} A$. Für
|
||||
eine Abbildung endlicher Mengen $\phi\colon D \to E$ bezeichne
|
||||
mit $\hat{\phi}\colon A^{E}\to A^{D}$ den von $A^{E} \to A$,
|
||||
$(a_e)_{e \in E} \mapsto a_{\phi(d)}$ für $d \in D$ induzierten $A$-Algebrahomomorphismus.
|
||||
Ein solches $\hat{\phi}$ ist endlich étale.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
Seien $A, B, C$ Ringe und $f\colon A \to B$, $g\colon A \to C$ total zerlegbar und $h\colon C \to B$ ein
|
||||
Ringhomomorphismus mit $f = hg$. Sei weiter $\mathfrak{p} \in \spec A$. Dann
|
||||
existiert ein $a \in A$, sodass $a \not\in \mathfrak{p}$ und $f, g$ und $h$ trivial über $D(a)$ sind. Das heißt
|
||||
es existieren endliche Mengen $D$ und $E$ und Isomorphismen $\alpha\colon A_a^{D} \to B_a$
|
||||
und $\beta\colon A_a^{E} \to C_a$ und eine Abbildung $\phi\colon D \to E$, sodass das Diagramm
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
B_a & & & \arrow[from=1-4,to=1-1,swap]{}{h} C_a \\
|
||||
& \arrow{ul}{\alpha} A_a^{D} & \arrow[swap]{l}{\hat{\phi}} A_a^{E} \arrow{ur}{\beta} & \\
|
||||
A_a \arrow[from=3-1,to=1-1]{}{f} \arrow{ur} & & & \arrow[from=3-4,to=3-1]{}{\operatorname{id}_{A_a}} \arrow{ul} A_a \arrow[from=3-4,to=1-4]{}{g}
|
||||
\end{tikzcd}
|
||||
\] kommutiert.
|
||||
\label{lemma:locally-trivial}
|
||||
\end{lemma}
|
||||
|
||||
\begin{satz}
|
||||
Seien $f\colon A \to B$ und $g\colon A \to C$ endlich étale Ringhomomorphismen und
|
||||
$h\colon C \to B$ ein Ringhomomorphismus mit $f = hg$. Dann ist $h$ endlich étale.
|
||||
\end{satz}
|
||||
\end{document}
|
||||
Binary file not shown.
Binary file not shown.
+490
-422
File diff suppressed because it is too large
Load Diff
Binary file not shown.
Binary file not shown.
Binary file not shown.
Binary file not shown.
@@ -0,0 +1,239 @@
|
||||
\documentclass[a4paper]{article}
|
||||
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage[T1]{fontenc}
|
||||
\usepackage{textcomp}
|
||||
\usepackage[german]{babel}
|
||||
\usepackage{amsmath, amssymb, amsthm}
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
|
||||
\title{Versuch einer axiomatischen Ethik}
|
||||
\author{Christian Merten}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\maketitle
|
||||
|
||||
\section{Einleitung}
|
||||
|
||||
%Die Philosophie ist eine hoffnungslose Wissenschaft.
|
||||
|
||||
Nach Moore ist eine Definition des Wortes \emph{gut} unmöglich. Gut
|
||||
wird von ihm als abstraktes, einfaches Grundwort vorausgesetzt, dessen Existenz er postuliert.
|
||||
Dieses Prinzip findet sich
|
||||
in der Mathematik, speziell der Mengenlehre wieder. Hier wird der Begriff der \emph{Menge}
|
||||
vorausgesetzt ohne diesen zu definieren. Es besteht jedoch ein wesentlicher Unterschied zwischen der
|
||||
Mengenlehre und der Ethik Moores: Die Mengenlehre befasst sich mit den Eigenschaften von
|
||||
Mengen, anhand vorgegebener Axiome. Moore hingegen schreibt gut keinerlei Eigenschaften
|
||||
zu, außer der charakterisierenden Eigenschaft des Guten, nicht definierbar zu sein.
|
||||
|
||||
Charakterisierungen als Definition zu missverstehen ist für Moore ein Fehlschluss, weshalb der
|
||||
hier anklingende Widerspruch nur oberflächlich ist.
|
||||
Eine Definition im Sinne von Moore ist eine analytische Aufsummierung
|
||||
der Bestandteile einer Sache. Dennoch charakterisiert Moore das Wort gut: Er postuliert es gebe
|
||||
nur ein einziges einfaches, nicht analytisch zerlegbares, also nicht definierbares Grundwort:
|
||||
Das Wort gut. Dadurch ist gut eindeutig charakterisiert, ohne eine Definition
|
||||
im Sinne von Moore zu geben.
|
||||
|
||||
Einen strengen Unterschied sieht Moore zwischen der Eigenschaft gut und dem \emph{Guten}.
|
||||
Letzteres bezeichne im Folgenden die Sammlung aller Gegenstände unseres Denkens, die die Eigenschaft
|
||||
gut tragen. Das Gute ist also ein analytisch zerlegter Begriff und somit das
|
||||
Spiegelbild von gut in der natürlichen Welt. Die grundlegende Aufgabe der Ethik ist
|
||||
diese Sammlung des Guten zu bestimmen.
|
||||
|
||||
Neben dem einzigen einfachen Grundwort gut gibt es in unserer Sprache auf den ersten
|
||||
Blick noch ein weiteres solches einfaches Grundwort: Das Adjektiv \emph{schlecht}. Das steht
|
||||
jedoch im Widerspruch zu Moores Postulat der Eindeutigkeit des Wortes gut. Der Fehler
|
||||
liegt darin, dass, der natürlichen, sprachlichen Intuition folgend, das Adjektiv schlecht,
|
||||
unter Voraussetzung der Existenz des Wortes gut, analytisch als das Gegenteil von
|
||||
gut definiert werden kann.
|
||||
|
||||
Dies führt zwingend zu folgender Beobachtung: Da die Bedeutung von gut nicht analytisch
|
||||
festlegbar ist, bleibt die Wahrheit des Satzes ,,Töten ist gut.`` unentscheidbar. Auf den ersten
|
||||
Blick werden die meisten Menschen diesen Satz als falsch bezeichnen. Für
|
||||
eine Person, die für sich das einzige, nicht analytisch zerlegbare Grundwort gut mit der
|
||||
der sprachlichen Intuition folgenden Bedeutung von schlecht assoziiert, wird diesen
|
||||
Satz jedoch als wahr ansehen.
|
||||
|
||||
Die Intuitionisten lösen dieses Dilemma, indem sie die Existenz eines weiteren
|
||||
Sinns postulieren: Die moralische Intuition. Diese detektiert die Eigenschaft
|
||||
gut und erlaubt somit eine Beantwortung der oben beschriebenen grundlegenden Frage
|
||||
der Ethik nach der Charakterisierung des Guten.
|
||||
|
||||
Dieser Aufsatz plädiert für eine andere Umgangsweise mit dem obigen Dilemma.
|
||||
|
||||
\section{Axiomatische Ethik}
|
||||
|
||||
Ein Axiom ist ein Grundsatz einer Theorie oder eines Systems, das innerhalb des Systems als wahr
|
||||
angenommen ist. Im Gegensatz zu einer These bedarf ein Axiom keines Beweises innerhalb der Theorie.
|
||||
Beispiele für axiomatische Wissenschaften sind
|
||||
die Mathematik, mit den Axiomen der Mengenlehre, oder die
|
||||
theoretische Physik der klassischen Mechanik, mit den Newtonschen Axiomen.
|
||||
|
||||
Ein bereits aufgetretenes ethisches Axiom ist Moores Postulat der Existenz und Eindeutigkeit
|
||||
des Grundworts gut. In diesem Sinne ist die Ethik Moores also bereits axiomatisch. Im Folgenden
|
||||
ist dieses Postulat die Grundannahme und es werden nur Axiomensysteme betrachtet, die dieses
|
||||
voraussetzen.
|
||||
|
||||
Der Ansatz zur Lösung des oben beschriebenen Dilemmas ist nun die Schaffung eines ausdrucksstarken
|
||||
Axiomensystems. Die Grundobjekte sind hier stets die Gegenstände unseres Denkens, über die moralische
|
||||
Urteile gefällt werden sollen. Ein mögliches Axiomensystem ist nun das folgende:
|
||||
|
||||
\begin{enumerate}[(i)]
|
||||
\item Jede nichtleere Sammlung von Gegenständen unseres Denkens enthält eine
|
||||
Sammlung als Element,
|
||||
die disjunkt zur ersten Sammlung ist.
|
||||
\item Das Gute ist eine nichtleere Sammlung.
|
||||
\item Das Schlechte ist eine nichtleere Sammlung.
|
||||
\item Ein Gegenstand unseres Denkens ist genau dann gut, wenn er nicht schlecht
|
||||
ist.
|
||||
\item Die Hintereinanderausführung zweier Handlungen ist genau dann gut,
|
||||
wenn beide Handlungen gut sind.
|
||||
\item Die Vereinigung von endlich vielen guten Gegenständen unseres
|
||||
Denkens zu einer Sammlung
|
||||
ist selbst wieder gut.
|
||||
\end{enumerate}
|
||||
|
||||
Einige dieser Axiome erscheinen dem Leser vermutlich naheliegend. Andere mögen auf
|
||||
den ersten Blick überraschend oder unklar wirken. Besonders das Axiom (i) benötigt
|
||||
eine Erläuterung: Da auch Sammlungen von Gegenständen unseres Denkens erneut ein Gegenstand
|
||||
unseres Denkens darstellen, können diese selbst wieder Elemente von solchen Sammlungen sein. Das
|
||||
erste Axiom verhindert nun, dass zyklische Verschachtelungen von Sammlungen existieren:
|
||||
Sei eine Sammlung gegeben. Dann betrachte man die
|
||||
Sammlung, die als einziges Element die erste Sammlung enthält. Die zweite Sammlung
|
||||
enthält dann nach Axiom (i) eine Sammlung, die disjunkt zu ersterer ist. Allerdings
|
||||
enthält die zweite Sammlung nur die erste Sammlung, also ist die zweite Sammlung disjunkt
|
||||
zur ersten Sammlung. In anderen Worten: Keine Sammlung enthält sich selbst als Element.
|
||||
|
||||
Hieraus lassen sich nun zahlreiche Eigenschaften des Guten ableiten. Beispielsweise
|
||||
ist das Gute selbst natürlich ein Gegenstand unseres Denkens. Es drängt sich also
|
||||
natürlicherweise die Frage auf, ob das Gute selbst gut ist. Mit anderen Worten:
|
||||
Ist das Gute im Guten enthalten? Die Erläuterung im letzten Absatz beantwortet die Frage eindeutig
|
||||
mit Nein, per Definition des Guten ist also das Gute nicht selbst gut.
|
||||
Das hat eine direkte Konsequenz: Nach Axiom (iv) ist das Gute schlecht.
|
||||
|
||||
Dieses Beispiel nährt Zweifel an einem Sinn für moralische Intuition. Die zunächst schwierig, wenn
|
||||
nicht sogar widersprüchlich
|
||||
klingende Aussage ,,Das Gute ist schlecht.`` hat in diesem axiomatischen Kontext nicht nur
|
||||
einen klaren Sinn, sie ist sogar zwingend wahr.
|
||||
|
||||
Die Schwierigkeit der Theorie ist, dass die Sammlung des Guten zwar nichtleer ist, jedoch
|
||||
die Axiome, außer dem Schlechten mit einem analogen Argument wie oben, keine Gegenstände
|
||||
unseres Denkens als gut charakterisiert.
|
||||
Per Axiom natürliche Objekte anzugeben, die gut sind, führt aber zwingend zu
|
||||
einem naturalistischen Fehlschluss. Es
|
||||
ist damit notwendigerweise gegeben, dass die Axiome ohne solche konkreten Angaben formuliert werden.
|
||||
|
||||
Damit
|
||||
ist das motivierende Dilemma folgendermaßen gelöst: Formale Eigenschaften des Wortes gut, werden
|
||||
durch Axiome festgehalten. Dadurch erhält das Wort gut eine über seine definierende
|
||||
Charakterisierung hinausgehende Bedeutung. Die Entscheidung, welche natürlichen Gegenstände
|
||||
unseres Denkens gut sind, findet dann zwingend im System und nicht außerhalb statt.
|
||||
|
||||
Hier liegt eine große Stärke dieses axiomatischen Ansatzes.
|
||||
Die Axiome geben lediglich einen formalen Rahmen für ethische Argumente. Sie sind
|
||||
somit unabhängig von den sozialen Standards, Werten und Normen einer Gesellschaft und
|
||||
können als universelle Grundlage für moralische Diskussionen angesehen werden.
|
||||
Moral ist dann eine Realisierung der Ethik, beispielsweise in einer Gesellschaft in
|
||||
der Werte und Normen die Sammlung des Guten mit Leben füllen. Selbstverständlich
|
||||
ist dabei nicht jedes Paar aus Gesellschaft und Werten eine Realisierung des obigen Axiomensystems.
|
||||
Die Aufgabe der Metaethik ist deshalb nun ein geeignetes Axiomensystem zu finden, das zum einen
|
||||
ausdrucksstark genug ist, um ethische Argumentationen auszuführen und andererseits hinreichend
|
||||
kompatibel ist mit realen Gesellschaften und Wertsystemen. Das heißt, dass sich genügend viele
|
||||
dieser letzten Systeme in dem axiomatischen Modell realisieren lassen.
|
||||
|
||||
In diesem Ansatz ist die Ethik die Wissenschaft, die aus den Axiomen ethische Thesen beweist.
|
||||
Die Metaethik lebt hier also eine Ebene höher als die Ethik und ist von dieser klar getrennt.
|
||||
Sie ist die Instanz, die über die Axiome streitet, blickt also von außen auf diese Grundsätze
|
||||
und ist somit überhaupt in der Lage über diese zu diskutieren. Wertvolle Kriterien sind
|
||||
Ausdrucksstärke und
|
||||
Widerspruchsfreiheit, wobei letztere bereits aus logischen Gründen nicht
|
||||
nachgewiesen werden kann\footnote{Siehe Gödel.}.
|
||||
Moores Ansatz ist ein leeres Axiomensystem, denn neben dem Postulat der Existenz des
|
||||
Grundbegriffs gut, fordert er keine weiteren Eigenschaften an die Eigenschaft gut. Sein
|
||||
System ist somit maximal ausdrucksschwach.
|
||||
|
||||
Das hier vorgeschlagene System ist klar ausdrucksstärker. Beispielsweise ist
|
||||
die folgende Aussage wahr: Seien zwei Handlungen gegeben.
|
||||
Falls eine der beiden schlecht ist, so ist zwingend die
|
||||
Hintereinanderausführung schlecht, denn angenommen diese wäre gut, so folgt mit Axiom (v),
|
||||
dass beide Handlungen gut sind.
|
||||
|
||||
Die Ausdrucksstärke macht das System natürlich auch streitbarer: Eine unmittelbare Konsequenz
|
||||
aus obigem Beispiel ist die folgende Aussage: Angenommen Stehlen ist schlecht. Dann ist der
|
||||
Diebstahl eines Kaugummis im lokalen Supermarkt, gefolgt von einer großzügigen Spende
|
||||
an ein Wiederaufbauprogramm in der Ukraine unabhängig vom Charakter der zweiten Handlung schlecht.
|
||||
Eine gute Handlung ist also eine vollkommen gute Handlung. Jeder Schritt muss alleine gesehen
|
||||
gut sein. In den meisten Gesellschaften mit ihren gängigen Wertsystemen wirkt das sicherlich, wie
|
||||
eine zweifelhafte Aussage. Das legt die Vermutung nahe, dass das hier beispielhaft
|
||||
angegebene Axiomensystem nicht hinreichend realisierbar ist. Vermutlich würden die meisten Menschen,
|
||||
die nicht gerade Putin sind, die obige Situation als ,,überwiegend gut`` bewerten.
|
||||
Moralische Urteile sind wie hier häufig relativ formuliert oder verwenden zumindest
|
||||
Abstufungen der Eigenschaft gut. Das steht natürlich im klaren Gegensatz zum vorliegenden Modell,
|
||||
das nur zwei Zustände zulässt: Ein Gegenstand unseres Denkens ist entweder gut oder schlecht.
|
||||
|
||||
Es ist wichtig darauf hinzuweisen, dass das obige Problem nicht bei Vereinigungen von
|
||||
Gegenständen unseres Denkens auftritt. Die Axiomatik ist hier entschieden schwächer, denn
|
||||
eine Vereinigung guter und schlechter Gegenstände kann durchaus gut sein. Ein klassisches
|
||||
Beispiel ist der Tyrannenmord. Während in den meisten Gesellschaften die Gegenstände
|
||||
,,Tyrannei abschaffen`` als gut und ,,Mord`` als schlecht eingestuft werden, wird dessen
|
||||
Vereinigung, Tyrannenmord, als gut eingeschätzt.
|
||||
|
||||
Bei genauer Betrachtung kann auch die Hinrichtung des Axioms (v)
|
||||
als nicht intuitiv erscheinen: Unter der
|
||||
in den meisten wissenschaftlichen Kreisen vertretenen Annahme,
|
||||
dass das Schützen des Klimas, beispielsweise durch das Entziehen von CO$_2$ aus der Atmosphäre,
|
||||
eine gute Handlung ist,
|
||||
folgt mit Axiom (v), dass auch die beliebig häufige Wiederholung dieser Handlung wieder eine gute
|
||||
Handlung ist. Jedoch führt das häufig wiederholte Entziehen von CO$_2$
|
||||
aus der Atmosphäre zwingend zu einer deutlichen Senkung der globalen Temperatur. Nach hinreichend
|
||||
häufiger Wiederholung ist also nicht mehr die globale Erwärmung das Problem, sondern die globale
|
||||
Abkühlung und die damit verbundene drohende menschengemachte Eiszeit. In
|
||||
dieser Situation wäre, zumindest gemäß gemäßigten wissenschaftlichen Kreisen, das Entziehen von
|
||||
CO$_2$ eine schlechte Handlung und das Verbrennen von Kohle die gute Handlung.
|
||||
Das sehr häufige Wiederholen der selben guten Handlung kann also eine schlechte Handlung sein.
|
||||
Wieder scheint also das vorgeschlagene Axiomensystem mit einem weit verbreiteten Wertesystem nicht
|
||||
kompatibel zu sein. Allerdings liegt diese Inkompatibilität nur an der Oberfläche. Der Fehler
|
||||
ist sprachliche Ungenauigkeit: Die Aussage ,,Entziehen von CO$_2$ ist gut`` ist eine fatal verkürzte
|
||||
Form von ,,Entziehen von CO$_2$ ist im Sommer 2022 gut``. Wenn also die Handlungen vollständig
|
||||
mit Ort und Zeit verstanden werden, dann lassen sich diese auch nur hintereinander
|
||||
ausführen, wenn die Zeiten miteinander kompatibel sind. Und dann lässt sich auch aus der
|
||||
Prämisse ,,Entziehen von CO$_2$ im Sommer 2022 ist gut``, nicht die Aussage
|
||||
,,Entziehen von CO$_2$ im Jahr 2100 ist gut`` ableiten.
|
||||
|
||||
\subsection{Endlichkeit}
|
||||
|
||||
Das Gute ist nicht endlich, denn wäre es endlich, so wäre es die endliche Vereinigung seiner
|
||||
guten Elemente, also nach Axiom (vi) selber gut, was im Widerspruch zu der Tatsache steht, dass
|
||||
das Gute schlecht ist. Hier wird also nicht nur postuliert, dass es überhaupt unendlich viele
|
||||
Gegenstände unseres Denkens gibt, was angesichts des Beispiels der Zahlen die meisten
|
||||
Menschen nicht verwundern wird, sondern auch, dass unendlich viele solche die Eigenschaft
|
||||
gut tragen. Das bedeutet, dass eine Moral, die dieses Axiomensystem realisiert, nicht
|
||||
durch eine endliche Auflistung guter Gegenstände unseres Denkens erklärt werden kann.
|
||||
Es drängt sich an dieser Stelle die Frage auf: Ist das Gute zumindest endlich erzeugt? Also
|
||||
gibt es eine endliche Sammlung elementarer guter Gegenstände unseres Denkens, sodass sich
|
||||
alle guten Dinge als Vereinigung oder Hintereinanderausführung dieser Elementaren ergeben? Ein
|
||||
Beispiel für eine Moral, die das Axiomensystem realisiert und endlich erzeugt ist, lässt
|
||||
sich im alten Testament finden: Die von den 10 Geboten ableitbare Moral. Das Gute ist dann
|
||||
gegeben als beliebige endliche Vereinigungen der Werte wie
|
||||
Ehrlichkeit, Treue oder Respekt des Eigentums, und beliebige Hintereinanderausführungen
|
||||
der so entstandenen guten Handlungen.
|
||||
|
||||
Vermutlich entstehen tatsächlich die meisten Moralsysteme aus einer endlichen Menge an Werten, in
|
||||
anderen Worten: Das Gute ist zumeist endlich erzeugt. Ein mögliches zusätzliches sinnvolles Axiom
|
||||
wäre also zu fordern, dass das Gute ein endliches Erzeugendensystem besitzt.
|
||||
|
||||
Die analoge Frage lässt sich natürlich auch für das Schlechte stellen. Das Axiomensystem lässt diese
|
||||
Frage jedoch offen, da keine Aussage über endliche Vereinigungen von schlechten Gegenständen
|
||||
unseres Denkens getroffen wird.
|
||||
|
||||
\section{Zusammenfassung}
|
||||
|
||||
Moores Postulat des eindeutigen einfachen Grundworts gut, zieht zwingend die Notwendigkeit nach sich,
|
||||
dieses Wort mit weiteren postulierten Eigenschaften zu versehen. Anstatt eine fast göttliche,
|
||||
übergeordnete Instanz zu fordern, die moralische Intuition, die unfehlbar entscheidet, was gut
|
||||
und schlecht ist, ist der hier geschilderte axiomatische Ansatz ein Versuch die formalen
|
||||
Eigenschaften des Guten festzulegen, um damit einen möglichst universellen Rahmen für
|
||||
Moralsysteme aller Art zu schaffen.
|
||||
|
||||
\end{document}
|
||||
@@ -0,0 +1,57 @@
|
||||
@book {hartshorne,
|
||||
AUTHOR = {Hartshorne, Robin},
|
||||
TITLE = {Residues and duality},
|
||||
SERIES = {Lecture Notes in Mathematics, No. 20},
|
||||
NOTE = {Lecture notes of a seminar on the work of A. Grothendieck,
|
||||
given at Harvard 1963/64,
|
||||
With an appendix by P. Deligne},
|
||||
PUBLISHER = {Springer-Verlag, Berlin-New York},
|
||||
YEAR = {1966},
|
||||
PAGES = {vii+423},
|
||||
MRCLASS = {14.55},
|
||||
MRNUMBER = {0222093},
|
||||
MRREVIEWER = {R. L. Knighten},
|
||||
}
|
||||
|
||||
@book {kashiwara,
|
||||
AUTHOR = {Kashiwara, Masaki and Schapira, Pierre},
|
||||
TITLE = {Sheaves on manifolds},
|
||||
SERIES = {Grundlehren der mathematischen Wissenschaften [Fundamental
|
||||
Principles of Mathematical Sciences]},
|
||||
VOLUME = {292},
|
||||
NOTE = {With a chapter in French by Christian Houzel,
|
||||
Corrected reprint of the 1990 original},
|
||||
PUBLISHER = {Springer-Verlag, Berlin},
|
||||
YEAR = {1994},
|
||||
PAGES = {x+512},
|
||||
ISBN = {3-540-51861-4},
|
||||
MRCLASS = {58G07 (18F20 32C38 35A27)},
|
||||
MRNUMBER = {1299726},
|
||||
}
|
||||
|
||||
@book {gelfand,
|
||||
AUTHOR = {Gelfand, Sergei I. and Manin, Yuri I.},
|
||||
TITLE = {Methods of homological algebra},
|
||||
SERIES = {Springer Monographs in Mathematics},
|
||||
EDITION = {Second},
|
||||
PUBLISHER = {Springer-Verlag, Berlin},
|
||||
YEAR = {2003},
|
||||
PAGES = {xx+372},
|
||||
ISBN = {3-540-43583-2},
|
||||
MRCLASS = {18-02 (18Exx 18Gxx 55U35)},
|
||||
MRNUMBER = {1950475},
|
||||
DOI = {10.1007/978-3-662-12492-5},
|
||||
URL = {https://doi.org/10.1007/978-3-662-12492-5},
|
||||
}
|
||||
|
||||
@article {mathew,
|
||||
AUTHOR = {Akhil Mathew},
|
||||
TITLE = {Verdier Duality},
|
||||
NOTE = {Expository Notes (version dated July 29, 2011), available at \url{https://math.uchicago.edu/~amathew/verd.pdf}},
|
||||
}
|
||||
|
||||
@article {mustata,
|
||||
AUTHOR = {Mircea Mustaţă},
|
||||
TITLE = {Soft sheaves on paracompact spaces and applications},
|
||||
NOTE = {Notes, available at \url{http://websites.umich.edu/~mmustata/SoftSheaves.pdf}},
|
||||
}
|
||||
Binary file not shown.
@@ -0,0 +1,565 @@
|
||||
\documentclass[a4paper]{../../notes}
|
||||
|
||||
\newcommand{\com}[1]{#1^{\text{\scalebox{0.7}{\textbullet}}}}
|
||||
\newcommand{\K}{\mathcal{K}}
|
||||
\renewcommand{\lim}{\varprojlim}
|
||||
\newcommand{\colim}[1]{\underset{#1}{\operatorname{colim}\;}}
|
||||
|
||||
\newcommand{\spec}{\operatorname{Spec }}
|
||||
|
||||
\newcommand{\sh}[1]{\mathcal{A}b(#1)}
|
||||
\newcommand{\supp}[1]{\operatorname{supp}(#1)}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\section{Overview}
|
||||
|
||||
These notes mostly follow \cite{mathew}. Some ideas are taken
|
||||
from \cite{gelfand}.
|
||||
|
||||
In the following, for a topological space $X$ denote by $\sh{X}$ the category
|
||||
of sheaves of abelian groups on $X$. Furthermore, denote by
|
||||
$\mathrm{D}^{+}(X)$ the bounded below derived category of $\sh{X}$.
|
||||
|
||||
\begin{definition}[Lower Shriek]
|
||||
Let $f\colon X \to Y$ be a continuous map of locally compact topological spaces.
|
||||
For $\mathcal{F} \in \sh{X}$
|
||||
and $U \subseteq Y$ open, let
|
||||
\[
|
||||
f_{!}(\mathcal{F})(U) = \{ s \in \mathcal{F}(f^{-1}(U)) \colon \supp{s} \xrightarrow{f} U \text{ proper}\}
|
||||
.\]
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}[Lower shriek of sheaf is a sheaf]
|
||||
Let $\mathcal{F} \in \sh{X}$ be a sheaf $f\colon X \to Y$ continuous.
|
||||
Then $f_{!}\mathcal{F}$ is a sheaf on $Y$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Clearly, $f_{!}\mathcal{F}$ is a sub-presheaf of the sheaf $f_{*} \mathcal{F}$. To show
|
||||
it is a sheaf, we need to verify that gluing sections in $f_{!}\mathcal{F}$ gives again a
|
||||
section in $f_{!}\mathcal{F}$.
|
||||
|
||||
Let $(U_i)_{i \in I}$ be a family of open sets in $Y$ and $s_i \in (f_{!} \mathcal{F})(U_i)$
|
||||
sections. Thus $s_i \in \mathcal{F}(f^{-1}(U_i))$ such that $\supp{s_i} \xrightarrow{f} U_i$
|
||||
is proper.
|
||||
Gluing yields a unique section $s \in \mathcal{F}(f^{-1}(U))$. We need
|
||||
to check that
|
||||
\[
|
||||
\supp{s} = \bigcup_{i \in I} \supp{s_i} \xlongrightarrow{f} \bigcup_{i \in I} U_i
|
||||
\] is proper. For this note that
|
||||
$\left(f|_{\supp{s}}\right)^{-1}(U_i) = f^{-1}(U_i) \cap \supp{s} = \supp{s_i}$ and
|
||||
being proper is local on the target.
|
||||
\end{proof}
|
||||
|
||||
The goal of this and the following talk is to prove the following theorem
|
||||
|
||||
\begin{theorem}[Verdier duality]
|
||||
If $X, Y$ are locally compact topological spaces of finite dimension,
|
||||
then $\mathrm{R}f_{!}$ admits a right adjoint
|
||||
$f^{!}\colon \mathrm{D}^{+}(Y) \to \mathrm{D}(X)$.
|
||||
\end{theorem}
|
||||
|
||||
To show the existence of the derivative of $f_{!}$, we need to introduce an adapted class of shaves.
|
||||
|
||||
\begin{definition}
|
||||
Let $X$ be a locally compact space, $\mathcal{F} \in \sh{X}$ and $Z \subseteq X$ a subset. Then
|
||||
define
|
||||
\[
|
||||
\mathcal{F}(Z) = \Gamma(Z, \mathcal{F}) = \Gamma(Z, i^{*}\mathcal{F})
|
||||
\] for $i\colon Z \to X$ the canonical inclusion.
|
||||
\end{definition}
|
||||
|
||||
\begin{bem}
|
||||
If $Z \subseteq X$ is a subset and $i\colon Z \to X$ the canonical inclusion, then
|
||||
\[
|
||||
\mathcal{F}(Z)
|
||||
=
|
||||
\left\{ (s_i, U_i)_{i \in I} \colon U_i \subseteq X \text{ open with } Z \subseteq \bigcup_{i \in I} U_i,
|
||||
s_i \in \mathcal{F}(U_i) \text{ with } (s_i)_z = (s_{j})_z \forall i, j \in I, z \in Z \cap U_i \cap U_j\right\} / \sim
|
||||
.\]
|
||||
where $(U_i, s_i)_{i \in I} \sim (V_j, t_j)_{j \in J}$
|
||||
if and only if $(s_i)_z = (t_j)_z$ for all $i \in I$, $j \in J$ and $z \in U_i \cap V_j \cap Z$.
|
||||
|
||||
For every open neighbourhood $U$ of $Z$, we have a restriction map
|
||||
\[
|
||||
\mathcal{F}(U) \to \mathcal{F}(Z), s \mapsto s|_Z \coloneqq [(s, U)]
|
||||
.\] This induces a map
|
||||
\[
|
||||
\colim{Z \subseteq U} \mathcal{F}(U)
|
||||
\to \mathcal{F}(Z)
|
||||
.\]
|
||||
\end{bem}
|
||||
|
||||
\begin{lemma}
|
||||
Let $X$ be a locally compact Hausdorff space and $\mathcal{F} \in \sh{X}$.
|
||||
If $Z \subseteq X$ is compact, the natural map
|
||||
\[
|
||||
\colim{Z \subseteq U} \mathcal{F}(U) \longrightarrow \mathcal{F}(Z)
|
||||
\] is an isomorphism.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Injectivity: Let $s \in \mathcal{F}(U)$ such that $s|_Z = 0$. Thus for all $z \in Z$,
|
||||
$s_z = 0$ and
|
||||
there exists an open neighbourhood
|
||||
$z \in U_z \subseteq U$ such that $s|_{U_z} = 0$. Thus $s|_{\bigcup U_z } = 0$. Since
|
||||
$Z \subseteq \bigcup_{z \in Z} U_z$, $s$ is zero in the colimit.
|
||||
|
||||
Surjectivity: Take $(s_i, U_i)_{i \in I} \in \mathcal{F}(Z)$. Thus
|
||||
$Z \subseteq \bigcup_{i \in I} U_i$ and by local compactness, for every $z \in Z$, there
|
||||
exists a compact neighbourhood $z \in K_z$ such that $K_z \subseteq U_{i_z}$ for
|
||||
some $i_z \in I$. Since $Z$ is compact, finitely many suffice, so we may assume
|
||||
$Z \subseteq \bigcup_{i=1}^{n} K_i$ and $K_i \subseteq U_i \subseteq X$.
|
||||
We now want to define a section on a neighbourhood of $Z$ that locally agrees with the $s_i$.
|
||||
|
||||
By induction, we may assume $n = 2$. By definition, $(s_1)_z = (s_2)_z$ for all $z \in Z \cap U_1 \cap U_2$,
|
||||
in particular $s_1|_{U_1 \cap U_2}$ and $s_2|_{U_1 \cap U_2}$ have the same restriction
|
||||
to $K_1 \cap K_2$. By the injectivity of the restriction map,
|
||||
there exists an open neighbourhood $K_1 \cap K_2 \subseteq V \subseteq U_1 \cap U_2$, such that
|
||||
$s_1|_V = s_2|_V$. Since $K_j \setminus V$ is closed in the compact $K_j$, for $j=1,2$
|
||||
the subset $K_j \setminus V$ is compact. Since $X$ is Hausdorff, there
|
||||
exist open neighbourhoods $K_j \setminus V \subseteq U_j' \subseteq U_j$ such that
|
||||
$U_1' \cap U_2' = \emptyset$. Now $s_1|_{U_1'}$, $s_2|_{U_2'}$ and
|
||||
$s_1|_V = s_2|_V$ glue to a section $w$ on $U_1' \cup U_2' \cup V \supseteq K_1 \cup K_2 \supseteq Z$
|
||||
such that $w|_Z = [(s_i, U_i)_{i \in I}]$.
|
||||
\end{proof}
|
||||
|
||||
\begin{definition}
|
||||
A sheaf $\mathcal{F} \in \sh{X}$ is \emph{soft} if
|
||||
$\mathcal{F}(X) \to \mathcal{F}(Z)$ is surjective whenever $Z \subseteq X$ is compact.
|
||||
\end{definition}
|
||||
|
||||
\begin{bem}
|
||||
In \cite{kashiwara} our notion of softness is called \emph{c-soft}.
|
||||
For $\sigma$-compact spaces the notions agree according to Exercise II.6 in \cite{kashiwara}.
|
||||
\end{bem}
|
||||
|
||||
\begin{bem}[Flasque sheaves are soft]
|
||||
Recall that a sheaf $\mathcal{F} \in \sh{X}$ is called \emph{flasque}, if
|
||||
for every open set $U \subseteq X$, the restriction map
|
||||
$\mathcal{F}(X) \to \mathcal{F}(U)$ is surjective. For $Z \subseteq X$ compact,
|
||||
we have a commutative diagram:
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\mathcal{F}(X) \arrow{rr} \arrow[twoheadrightarrow]{dr} & & \mathcal{F}(Z) \\
|
||||
& \colim{Z \subseteq U} \mathcal{F}(U) \arrow{ur}{\simeq} &
|
||||
\end{tikzcd}
|
||||
.\] Thus $\mathcal{F}$ is soft.
|
||||
\end{bem}
|
||||
|
||||
\begin{satz}
|
||||
Let $X$ be a locally compact topological space.
|
||||
If $\mathcal{F} \in \sh{X}$ is soft, $K \subseteq X$ is compact and $K \subseteq U$ is an open neighbourhood,
|
||||
any section over $K$ can be extended to a global section with compact support contained in $U$.
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
Let $s \in \mathcal{F}(K)$.
|
||||
By local compactness, there exists a compact neighbourhood $L$ of $K$ with $L \subseteq U$. Then
|
||||
$K \cap \partial L = \emptyset$. Consider the section on $K \cup \partial L$ given by
|
||||
$s$ on $K$ and zero on $\partial L$. Since $\mathcal{F}$ is soft, this can be extended
|
||||
to a global section, and a fortiori to a section $t$ over $L$. Now
|
||||
the sections given by $t$ on $L$ and $0$ on $\overline{X \setminus L}$ glue to a compactly
|
||||
supported extension of $s$. Since $L \subseteq U$, its support is contained in $U$.
|
||||
\end{proof}
|
||||
|
||||
\subsection{Compactly supported cohomology}
|
||||
|
||||
Let $X$ be a topological space.
|
||||
|
||||
\begin{bem}[Support]
|
||||
For $\mathcal{F} \in \sh{X}$, $U \subseteq X$ open and a section $s \in \mathcal{F}(U)$,
|
||||
its support $\supp{s}$ is defined as
|
||||
\[
|
||||
\{ x \in U\colon s_x \neq 0\}
|
||||
.\] This set is always closed, as its complement is open.
|
||||
\end{bem}
|
||||
|
||||
\begin{definition}
|
||||
Let $U \subseteq X$ be open and $\mathcal{F} \in \sh{X}$. We define
|
||||
$\Gamma_c(U, \mathcal{F})$ as the subgroup of $\Gamma(U, \mathcal{F})$ consisting of
|
||||
sections with compact support.
|
||||
\end{definition}
|
||||
|
||||
\begin{bem}
|
||||
If $s, t \in \Gamma(U, \mathcal{F})$ have compact support, so does $s + t$. Thus
|
||||
$\Gamma_c(U, \mathcal{F})$ is indeed a subgroup of $\Gamma(U, \mathcal{F})$.
|
||||
|
||||
Taking $U = X$, this defines a functor $\Gamma_c = \Gamma_c(X, \cdot)\colon \sh{X} \to \mathcal{A}b$
|
||||
\end{bem}
|
||||
|
||||
\begin{bem}[Lower shriek and compact support]
|
||||
Let $f\colon X \to \{ *\} $ be the unique continuous map from $X$ to the one point space.
|
||||
Then $f_{!} \cdot = \Gamma_c(X, \cdot)$
|
||||
\end{bem}
|
||||
|
||||
\begin{satz}
|
||||
$\Gamma_c$ is left exact.
|
||||
\label{satz:gamma_c-left-exact}
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}''$ be an exact sequence
|
||||
in $\sh{X}$. This induces a commutative diagram
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & \Gamma(X, \mathcal{F}') \arrow{r}
|
||||
& \Gamma(X, \mathcal{F}) \arrow{r}
|
||||
& \Gamma(X, \mathcal{F}'') \\
|
||||
0 \arrow{r} & \Gamma_c(X, \mathcal{F}') \arrow{r} \arrow[hookrightarrow]{u}
|
||||
& \Gamma_c(X, \mathcal{F}) \arrow{r} \arrow[hookrightarrow]{u}
|
||||
& \Gamma_c(X, \mathcal{F}'') \arrow[hookrightarrow]{u}
|
||||
\end{tikzcd}
|
||||
,\] where the first row is exact. Since the vertical arrows are inclusions,
|
||||
the injectivity of $\Gamma_c(X, \mathcal{F}') \to \Gamma_c(X, \mathcal{F})$ is immediate. Let now
|
||||
$s \in \Gamma_c(X, \mathcal{F}) \subseteq \Gamma(X, \mathcal{F})$
|
||||
such that $s$ becomes zero in $\Gamma_c(X, \mathcal{F}'')$. Thus
|
||||
by exactness of the first row, $s \in \Gamma(X, \mathcal{F}')$. Since $s \in \Gamma_c(X, \mathcal{F})$,
|
||||
$s$ is compactly supported, so $s \in \Gamma_c(X, \mathcal{F}')$.
|
||||
\end{proof}
|
||||
|
||||
\begin{satz}
|
||||
Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ be an exact sequence
|
||||
in $\sh{X}$. Suppose $\mathcal{F}'$ is soft. Then the sequence
|
||||
$0 \to \Gamma_c(X, \mathcal{F}') \to \Gamma_c(X, \mathcal{F}) \to \Gamma_c(X, \mathcal{F}'') \to 0$
|
||||
is also exact.
|
||||
|
||||
\label{satz:soft-gamma_c-exact}
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
By \ref{satz:gamma_c-left-exact}, we only need to show surjectivity on the right.
|
||||
|
||||
Suppose first that $X$ is compact and let $s \in \Gamma_c(X, \mathcal{F}'') = \Gamma(X, \mathcal{F}'')$.
|
||||
Since $\mathcal{F} \to \mathcal{F}'' \to 0$ is exact, there exist
|
||||
a covering $X = \bigcup_{i \in I} U_i$ and lifts $t_i \in \mathcal{F}(U_i)$
|
||||
of $s|_{U_i}$. By local compactness of $X$, we may assume, after a possible refinement, that each
|
||||
$U_i$ contains a compact set $V_i$ whose interiors still cover $X$. Since
|
||||
$X$ is compact, we may assume $I$ is finite. To piece together the $t_i$, we may assume, by induction,
|
||||
that $\#I = 2$.
|
||||
|
||||
Consider $t_1|_{U_1 \cap U_2} - t_2|_{U_1 \cap U_2}$. This is necessarily a section $e'$ of
|
||||
$\mathcal{F}'(U_1 \cap U_2)$ as it maps to zero in $\mathcal{F}''(U_1 \cap U_2)$. Restricting
|
||||
$e'$ to the compact $V_1 \cap V_2$ and extending it by softness, yields a global section $e$ of
|
||||
$\mathcal{F}'$. Now
|
||||
\[
|
||||
(t_2|_{V_2} + e|_{V_2})|_{V_1 \cap V_2} = t_2|_{V_1 \cap V_2} + e'|_{V_1 \cap V_2} = t_1|_{V_1 \cap V_2}
|
||||
.\] Thus $t_1|_{V_1}, t_2|_{V_2} + e|_{V_2}$ glue to a global section $t$ of $\mathcal{F}$
|
||||
with image $s$.
|
||||
|
||||
Now for general $X$: Let $s \in \mathcal{F}''(X)$ with compact support $Z$. By local compactness,
|
||||
there exists a compact neighbourhood $Z' \subseteq X$ of $Z$. Since
|
||||
pullback of sheaves is exact and restriction of soft sheaves to closed subsets preserves softness,
|
||||
applying the result to $Z'$,
|
||||
yields a section $t' \in \mathcal{F}(Z')$ lifting $s|_{Z'}$. The restriction
|
||||
$t'|_{\partial Z'}$ maps to $s|_{\partial Z'} = 0$, so $t'|_{\partial Z'} \in \mathcal{F}'(\partial Z')$.
|
||||
Since $\partial Z'$ is compact and $\mathcal{F}'$ is soft, $t'|_{\partial Z'}$
|
||||
extends to a global section $b$ of $\mathcal{F}'$. Thus
|
||||
\[
|
||||
(t' - b|_{Z'})|_{\partial Z'} = t'|_{\partial Z'} - t'|_{\partial Z'} = 0
|
||||
.\] So
|
||||
$t' - b|_{Z'}$ on $Z'$ and $0$ on $\overline{X \setminus Z'}$ glue to a global section
|
||||
$t$ of $\mathcal{F}$. Then $t|_{Z'} = t' - b|_{Z'}$ maps to $s|_{Z'}$ since
|
||||
$b \in \mathcal{F}'(X)$. Since $\supp{t}, \supp{s} \subseteq Z'$, $t$ is a compactly supported lift of $s$.
|
||||
\end{proof}
|
||||
|
||||
\begin{korollar}
|
||||
If $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ is an exact sequence
|
||||
in $\sh{X}$ and $\mathcal{F}', \mathcal{F}$ are soft, then
|
||||
$\mathcal{F}''$ is soft too.
|
||||
\label{kor:soft-2+3}
|
||||
\end{korollar}
|
||||
|
||||
\begin{proof}
|
||||
Let $Z \subseteq X$ be compact.
|
||||
Since restricting to a closed subset is exact and preserves softness,
|
||||
by \ref{satz:soft-gamma_c-exact} $\Gamma_c(Z, \mathcal{F}) \to \Gamma_c(Z, \mathcal{F}'')$ is surjective.
|
||||
This yields a commutative
|
||||
diagram
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\Gamma_c(X, \mathcal{F}) \arrow[twoheadrightarrow]{d} \arrow{r} & \Gamma_c(X, \mathcal{F}'')
|
||||
\arrow{d} \\
|
||||
\Gamma_c(Z, \mathcal{F}) \arrow[twoheadrightarrow]{r} & \Gamma_c(Z, \mathcal{F}'')
|
||||
\end{tikzcd}
|
||||
,\] where the left vertical arrow is surjective, since $\mathcal{F}$ is soft. Since
|
||||
the composition is surjective, $\Gamma_c(X, \mathcal{F}'') \to \Gamma_c(Z, \mathcal{F}'')$ is also
|
||||
surjective.
|
||||
\end{proof}
|
||||
|
||||
\begin{korollar}
|
||||
Soft sheaves are $\Gamma_c$-acyclic.
|
||||
\label{kor:soft-gamma_c-acyclic}
|
||||
\end{korollar}
|
||||
|
||||
\begin{proof}
|
||||
Let $\mathcal{F} \in \sh{X}$ be soft and
|
||||
embed $\mathcal{F}$ in an injective sheaf $\mathcal{I}$. This yields an exact sequence
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & \mathcal{F} \arrow{r}
|
||||
& \mathcal{I} \arrow{r}
|
||||
& \mathcal{G} \arrow{r}
|
||||
& 0
|
||||
\end{tikzcd}
|
||||
.\]
|
||||
Since $\mathcal{I}$ is injective, in particular flasque, hence soft,
|
||||
by \ref{kor:soft-2+3}, $\mathcal{G}$ is soft.
|
||||
We proceed by induction. For $i = 1$ consider the exact sequence
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & \Gamma_c(X, \mathcal{F}) \arrow{r}
|
||||
& \Gamma_c(X, \mathcal{I}) \arrow{r}
|
||||
& \Gamma_c(X, \mathcal{G}) \arrow{r}
|
||||
& H_c^{1}(X, \mathcal{F}) \arrow{r}
|
||||
& \underbrace{H_c^{1}(X, \mathcal{I})}_{= 0}
|
||||
\end{tikzcd}
|
||||
.\]
|
||||
Since $\mathcal{F}$ is soft, $\Gamma_c(X, \mathcal{I}) \to \Gamma_c(X, \mathcal{G})$ is
|
||||
surjective. By the exactness of the sequence, $H_c^{1}(X, \mathcal{F})$ vanishes.
|
||||
Now assume $H_c^{i}(X, \mathcal{F}) = 0$ for any soft sheaf $\mathcal{F}$. Then the exact sequence
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\underbrace{H_c^{i}(X, \mathcal{I})}_{= 0} \arrow{r} & H_c^{i}(X, \mathcal{G}) \arrow{r}
|
||||
& H_c^{i+1}(X, \mathcal{F}) \arrow{r}
|
||||
& \underbrace{H_c^{i+1}(X, \mathcal{I})}_{= 0}
|
||||
\end{tikzcd}
|
||||
\] yields an isomorphism $H_c^{i}(X, \mathcal{G}) \simeq H_c^{i+1}(X, \mathcal{F})$ and
|
||||
since $\mathcal{G}$ is soft, the left hand side is zero by induction hypothesis.
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}
|
||||
Let $f\colon X \to Y$ be a continuous map of locally compact topological spaces. If $Y$ is Hausdorff and
|
||||
$\mathcal{F} \in \sh{X}$, then there is a natural isomorphism
|
||||
\[
|
||||
(R^{i}f_{!}\mathcal{F})_y \simeq H_c^{i}(f^{-1}(y), \mathcal{F}|_{f^{-1}(y)})
|
||||
\] for each $y \in Y$.
|
||||
\label{thm:base-change}
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
Denote by $X_y$ the fibre of $f$ over $y$ and by $\mathcal{F}$ the restriction to $X_y$.
|
||||
Let $y \in Y$. Since $R^{i}f_{!}$ is a derived functor, it is a universal $\delta$-functor. Since restriction
|
||||
of soft sheaves to closed subspaces preserves softness, the $\delta$-functor
|
||||
$\mathcal{F} \mapsto H_c^{i}(X_y, \mathcal{F}_y)$ vanishes for soft sheaves and $i > 0$. Thus
|
||||
it is effaceable and hence universal. Therefore it suffices to define a natural isomorphism
|
||||
in degree $0$.
|
||||
|
||||
Let $y \in U \subseteq Y$ open. Then consider the natural map
|
||||
\begin{salign*}
|
||||
(f_{!}\mathcal{F})(U) &\longrightarrow \Gamma_c(X_y, \mathcal{F}_y) \\
|
||||
s &\longmapsto s|_{X_y}
|
||||
.\end{salign*}
|
||||
This is well-defined, since for any $s \in \mathcal{F}(f^{-1}(U))$ with
|
||||
$\supp{s} \xrightarrow{f} U$ proper, we have
|
||||
\[
|
||||
\supp{s|_{X_y}} = \supp{s} \cap X_y = \left( f|_{\supp{s}}^{U} \right)^{-1}(y)
|
||||
\] and the right hand side is compact. This map induces
|
||||
a natural map
|
||||
\[
|
||||
(f_{!}\mathcal{F})_y = \colim{y \in U \subseteq Y} (f_{!}\mathcal{F})(U)
|
||||
\longrightarrow \Gamma_c(X_y, \mathcal{F}_y)
|
||||
.\]
|
||||
|
||||
Injectivity: Let $s \in (f_{!}\mathcal{F})(U)$ such that $s|_{X_y} = 0$. Thus
|
||||
$s \in \mathcal{F}(f^{-1}(U))$ and $\supp{s} \xrightarrow{f} U$ is proper. Since
|
||||
$s|_{X_y} = 0$, $f^{-1}(y) \cap \supp{s} = X_y \cap \supp{s} = \emptyset$, in particular
|
||||
$y \not\in f(\supp{s})$. Let $y \in U'$ be the complement of $f(\supp{s})$ in $U$.
|
||||
Since $\supp{s} \xrightarrow{f} U$ is proper, $f(\supp{s})$ is closed in $U$, so
|
||||
$U'$ is open in $U$ and hence in $Y$. Moreover
|
||||
\[
|
||||
f^{-1}(U') \cap \supp{s}
|
||||
\subseteq f^{-1}(U') \cap f^{-1}(f(\supp{s}))
|
||||
= f^{-1}(U' \cap f(\supp{s}))
|
||||
= f^{-1}(\emptyset)
|
||||
= \emptyset
|
||||
.\]
|
||||
Hence $s|_{f^{-1}(U')} = 0$, so $s|_{U'} = 0$.
|
||||
|
||||
Surjectivity: Suppose first $\mathcal{F}$ is soft and let
|
||||
$s \in \Gamma_c(X_y, \mathcal{F}_y)$. Since $\mathcal{F}$ is soft, we may extend
|
||||
$s \in \mathcal{F}(X_y)$ to a compactly supported $s \in \mathcal{F}(X) = (f_{*}\mathcal{F})(Y)$.
|
||||
Since $Y$ is Hausdorff, every compact $K \subseteq Y$ is closed and therefore its preimage
|
||||
under $f|_{\supp{s}}$ is closed in the compact $\supp{s}$, thus itself compact. Hence
|
||||
$f|_{\supp{s}}\colon \supp{s} \to Y$ is proper and $s \in (f_{!}\mathcal{F})(Y)$.
|
||||
|
||||
For arbitrary $\mathcal{F}$, there exists an exact sequence
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & \mathcal{F} \arrow{r}
|
||||
& \mathcal{I} \arrow{r}
|
||||
& \mathcal{J}
|
||||
\end{tikzcd}
|
||||
\] with $\mathcal{I}, \mathcal{J}$ soft (e.g. injective). The functors
|
||||
$(f_{!} \cdot )_y$ and $\Gamma_c(X_y, \cdot |_{X_y})$ are left exact, so we have a commuting diagram
|
||||
with exact rows:
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & (f_!\mathcal{F})_y \arrow{r} \arrow{d}
|
||||
& (f_!\mathcal{I})_y \arrow{r} \arrow{d}{\simeq}
|
||||
& (f_!\mathcal{J})_y \arrow{d}{\simeq} \\
|
||||
0 \arrow{r} & \Gamma_c(X_y, \mathcal{F}_y) \arrow{r}
|
||||
& \Gamma_c(X_y, \mathcal{I}_y) \arrow{r}
|
||||
& \Gamma_c(X_y, \mathcal{J}_y)
|
||||
\end{tikzcd}
|
||||
.\] The five-lemma yields the desired isomorphism.
|
||||
\end{proof}
|
||||
|
||||
\begin{theorem}
|
||||
Consider a cartesian diagram of locally compact Hausdorff spaces:
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
X \times_Y Z \arrow{r}{f'} \arrow{d}{p'} & X \arrow{d}{p} \\
|
||||
Z \arrow{r}{f} & Y
|
||||
\end{tikzcd}
|
||||
.\] Then there is a natural isomorphism, for any
|
||||
$\com{\mathcal{F}} \in \mathcal{D}^{+}(X)$,
|
||||
\[
|
||||
f^{*} \mathrm{R}p_{!} \com{\mathcal{F}} \simeq \mathrm{R}p_!' f'^{*} \com{\mathcal{F}}
|
||||
.\]
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
By the universal property of derived functors, it suffices to define a natural transformation
|
||||
$f^{*}p_{!} \to \mathrm{R} p_{!}'f'^{*}$. By composing with the canonical
|
||||
natural transformation $p_{!}'f'^{*} \to \mathrm{R}p_{!}'f'^{*}$, it suffices to define
|
||||
the dotted arrow in the diagram below
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
f^{*}p_{!} \arrow[dashed]{rr} \arrow[dotted]{dr} & & \mathrm{R} p_{!}'f'^{*} \\
|
||||
& p_{!}'f'^{*} \arrow[swap]{ur}{can} &
|
||||
\end{tikzcd}
|
||||
.\] By naturality, it is sufficient to define for $\mathcal{G} \in \sh{X}$ a natural map
|
||||
$f^{*}p_! \mathcal{G} \to p_!'f'^{*}\mathcal{G}$. Since
|
||||
$f^{*} \dashv f_{*}$, this is equivalent to defining a natural map
|
||||
$p_!\mathcal{G} \to f_{*} p_{!}'f'^{*} \mathcal{G}$.
|
||||
|
||||
Again using $f'^{*} \dashv f'_{*}$, the map $\text{id}_{f'^{*} \mathcal{G}}$ induces a map
|
||||
$\mathcal{G} \to f'_{*} f'^{*} \mathcal{G}$. Applying
|
||||
$p_{*}$ yields $p_{*} \mathcal{G} \to p_{*}f'_{*}f'^{*} \mathcal{G}$. By the commutativity of the diagram
|
||||
we have $p_{*} f'_{*} = (pf')_{*} = (fp')_{*} = f_{*} p'_{*}$, so a map
|
||||
$\varphi\colon p_{*} \mathcal{G} \to f_{*} p'_{*} f'^{*} \mathcal{G}$.
|
||||
|
||||
For $U \subseteq Y$ open, this induces a map
|
||||
\[
|
||||
\varphi_U\colon \mathcal{G}(p^{-1}(U)) \longrightarrow (f'^{*} \mathcal{G})(p'^{-1}(f^{-1}(U)))
|
||||
.\]
|
||||
Let now $s \in \mathcal{G}(p^{-1}(U))$ such that
|
||||
$\supp{s} \xrightarrow{p} U$ is proper. Since $f'^{*}$ preserves stalks, for
|
||||
$(x, z) \in p^{-1}(U) \times_U f^{-1}(U)$ we have the following equivalences
|
||||
\[
|
||||
(x, z) \in \supp{\varphi_U(s)}
|
||||
\iff \varphi_U(s)_{(x, z)} \neq 0
|
||||
\iff s_{f'(x,z)} \neq 0
|
||||
\iff s_{x} \neq 0
|
||||
\iff x \in \supp{s}
|
||||
.\] Thus $\supp{\varphi_U(s)} = \supp{s} \times_{U} f^{-1}(U)$. We therefore have the following
|
||||
commutative diagram:
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\supp{s} \times_{U} f^{-1}(U) \arrow{d} \arrow{r} & \supp{s} \arrow{d} \\
|
||||
f^{-1}(U) \arrow{r} & U
|
||||
\end{tikzcd}
|
||||
.\] By assumption the right vertical arrow is proper. Since properness is stable under (topological)
|
||||
base change, the left vertical arrow is proper too. Hence
|
||||
$\supp{\varphi_U(s)} \xrightarrow{p'} f^{-1}(U)$ is proper and
|
||||
\[
|
||||
\varphi_U(s) \in (p'_{!}f'^{*} \mathcal{G})(f^{-1}(U)) = (f_{*} p'_{!}f'^{*} \mathcal{G})(U)
|
||||
.\] Thus $\varphi$ restricts to a natural map
|
||||
\[
|
||||
p_{!} \mathcal{G} \longrightarrow f_{*} p'_{!} f'^{*} \mathcal{G}
|
||||
.\]
|
||||
|
||||
To check that this is an isomorphism, we can use the fact that both functors are
|
||||
way-out functors in the sense of Section 7 in \cite{hartshorne}. Thus we only need to check
|
||||
this for a single sheaf $\mathcal{F} \in \sh{X}$, i.e. we want to show
|
||||
\[
|
||||
f^{*} R^{i} p_{!} \mathcal{F} \xlongrightarrow{\simeq} R^{i}p_{!}'f'^{*}\mathcal{F}
|
||||
\] for all $i \ge 0$. Again by universality of the $\delta$-functors involved,
|
||||
we may assume $i = 0$. Moreover, we can check this at the level of stalks. Let $z \in Z$. Then
|
||||
on the left hand side
|
||||
\begin{equation}
|
||||
(f^{*}p_{!}\mathcal{F})_z
|
||||
\simeq
|
||||
(p_{!} \mathcal{F})_{f(z)}
|
||||
\stackrel{\ref{thm:base-change}}{\simeq}
|
||||
\Gamma_c(p^{-1}(f(z)), \mathcal{F}|_{p^{-1}(f(z))})
|
||||
=
|
||||
\Gamma_c(f'(p'^{-1}(z))), \mathcal{F}|_{f'(p'^{-1}(z))})
|
||||
\label{eq:1}
|
||||
\end{equation}
|
||||
On the right hand side, we have
|
||||
\begin{equation}
|
||||
(p'_{!} f'^{*} \mathcal{F})_z
|
||||
\stackrel{\ref{thm:base-change}}{\simeq}
|
||||
\Gamma_c(p'^{-1}(z), (f'^{*} \mathcal{F})|_{p'^{-1}(z)})
|
||||
\label{eq:2}
|
||||
\end{equation}
|
||||
$\mathcal{F}|_{f'(p'^{-1}(z))}$ and
|
||||
$(f'^{*} \mathcal{F})|_{p'^{-1}(z)}$ are given as the sheafification of the same presheaf, indeed:
|
||||
\begin{salign*}
|
||||
\colim{p'^{-1}(z) \subseteq U \subseteq X \times_Y Z} \; (f'^{*}\mathcal{F})(U)
|
||||
&= \colim{p'^{-1}(z) \subseteq U \subseteq X \times_Y Z} \quad
|
||||
\colim{f'(U) \subseteq V \subseteq X} \; \mathcal{F}(V) \\
|
||||
&= \colim{f'(p'^{-1}(z)) \subseteq V \subseteq X} \; \mathcal{F}(V)
|
||||
.\end{salign*}
|
||||
This shows (\refeq{eq:1}) $\simeq$ (\refeq{eq:2}) and concludes the proof.
|
||||
\end{proof}
|
||||
|
||||
\begin{satz}
|
||||
Soft sheaves are $f_!$-acyclic. In particular, if
|
||||
$0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ is an exact sequence in $\sh{X}$
|
||||
and $\mathcal{F}'$ is soft, then the sequence
|
||||
$0 \to f_!\mathcal{F}' \to f_!\mathcal{F} \to f_!\mathcal{F}'' \to 0$ is exact.
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
Let $i > 0$ and $\mathcal{F} \in \sh{X}$ be soft. Then for $y \in Y$
|
||||
\begin{salign*}
|
||||
(R^{i}f_!\mathcal{F})_y
|
||||
\stackrel{\ref{thm:base-change}}{\simeq} H_c^{i}(f^{-1}(y), \mathcal{F}|_{f^{-1}(y)})
|
||||
\; \stackrel{\ref{kor:soft-gamma_c-acyclic}}{=} \; 0
|
||||
,\end{salign*}
|
||||
since the restriction of a soft sheaf to a closed subset is soft.
|
||||
\end{proof}
|
||||
|
||||
\begin{bsp}
|
||||
Let $U \subseteq X$ be open and $j\colon U \to X$ the inclusion map. By looking at stalks,
|
||||
one finds that $j_!\mathcal{F}$ for $\mathcal{F} \in \sh{U}$ is just extension by zero.
|
||||
\end{bsp}
|
||||
|
||||
\begin{satz}[Lower shriek preserves softness]
|
||||
If $f\colon X \to Y$ is continuous and $\mathcal{F} \in \sh{X}$ is soft, then
|
||||
$f_! \mathcal{F}$ is soft too.
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
Let $Z \subseteq Y$ be compact and
|
||||
$s \in (f_!\mathcal{F})(Z) \simeq \colim{Z \subseteq U \subseteq Y} (f_!\mathcal{F})(U)$. Then
|
||||
there exists an open neighbourhood $U$ of $Z$ and an extension
|
||||
$\tilde{s} \in (f_!\mathcal{F})(U) \subseteq \mathcal{F}(f^{-1}(U))$ with
|
||||
$\supp{\tilde{s}} \xrightarrow{f} U$ proper. Since $Y$ is locally compact, there exists
|
||||
a compact neighbourhood $L \subseteq U$ of $Z$. Restricting $\tilde{s}$ to the compact
|
||||
$K \coloneqq \left(f|_{\supp{\tilde{s}}}\right)^{-1}(L) \subseteq \supp{\tilde{s}}$
|
||||
and extending by softness of $\mathcal{F}$, yields a compactly supported global section
|
||||
$t \in \mathcal{F}(X) = (f_{*}\mathcal{F})(Y)$ such that $t|_Z = s$. Since
|
||||
$\supp{t}$ is compact and $Y$ is Hausdorff, $\supp{t} \xrightarrow{f} Y$ is proper.
|
||||
\end{proof}
|
||||
|
||||
\begin{korollar}[Leray spectral sequence]
|
||||
Given maps $f\colon X \to Y$, $g\colon Y \to Z$ of locally compact Hausdorff spaces,
|
||||
there is a natural isomorphism
|
||||
$\mathrm{R}(g \circ f)_{!} \simeq \mathrm{R}g_{!} \circ \mathrm{R}f_{!}$.
|
||||
\end{korollar}
|
||||
|
||||
\begin{proof}
|
||||
Since soft sheaves are $f_{!}$ (and $g_!$) acyclic and $f_{!}$ maps
|
||||
soft sheaves to soft sheaves, the result follows from
|
||||
Proposition 5.4 in \cite{hartshorne}.
|
||||
\end{proof}
|
||||
|
||||
\bibliographystyle{alpha}
|
||||
\bibliography{refs}
|
||||
|
||||
\end{document}
|
||||
Binary file not shown.
@@ -0,0 +1,682 @@
|
||||
\documentclass{../../notes}
|
||||
|
||||
\newcommand{\com}[1]{#1^{\text{\scalebox{0.7}{\textbullet}}}}
|
||||
\newcommand{\K}{\mathcal{K}}
|
||||
\renewcommand{\lim}{\varprojlim}
|
||||
\newcommand{\colim}[1]{\underset{#1}{\operatorname{colim}\;}}
|
||||
|
||||
\newcommand{\spec}{\operatorname{Spec }}
|
||||
|
||||
\newcommand{\sh}[1]{\mathcal{A}b(#1)}
|
||||
\newcommand{\supp}[1]{\operatorname{supp}(#1)}
|
||||
|
||||
\title{Lower shriek}
|
||||
\author{Christian Merten}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\section{Preliminaries}
|
||||
|
||||
These notes mostly follow \cite{mathew}. Some ideas are taken
|
||||
from \cite{gelfand} and \cite{kashiwara}.
|
||||
|
||||
In the following, a topological space $X$ is always assumed to be locally compact and Hausdorff.
|
||||
Denote by $\sh{X}$ the category
|
||||
of sheaves of abelian groups on $X$. Furthermore, denote by
|
||||
$\mathrm{D}(X)$ (respectively $\mathrm{D}^{+}(X)$) the (bounded below) derived category of $\sh{X}$.
|
||||
|
||||
\begin{definition}[Lower Shriek]
|
||||
Let $f\colon X \to Y$ be a continuous map of spaces.
|
||||
For $\mathcal{F} \in \sh{X}$
|
||||
and $U \subseteq Y$ open, let
|
||||
\[
|
||||
f_{!}(\mathcal{F})(U) = \{ s \in \mathcal{F}(f^{-1}(U)) \colon \supp{s} \xrightarrow{f} U \text{ proper}\}
|
||||
.\]
|
||||
\end{definition}
|
||||
|
||||
\begin{bem}[Support]
|
||||
For $\mathcal{F} \in \sh{X}$, $U \subseteq X$ open and a section $s \in \mathcal{F}(U)$,
|
||||
its support $\supp{s}$ is defined as
|
||||
\[
|
||||
\{ x \in U\colon s_x \neq 0\}
|
||||
.\] This set is always closed, as its complement is open.
|
||||
\end{bem}
|
||||
|
||||
\color{gray}
|
||||
|
||||
\begin{lemma}[Lower shriek of sheaf is a sheaf]
|
||||
Let $\mathcal{F} \in \sh{X}$ be a sheaf $f\colon X \to Y$ continuous.
|
||||
Then $f_{!}\mathcal{F}$ is a sheaf on $Y$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Clearly, $f_{!}\mathcal{F}$ is a sub-presheaf of the sheaf $f_{*} \mathcal{F}$. To show
|
||||
it is a sheaf, we need to verify that gluing sections in $f_{!}\mathcal{F}$ gives again a
|
||||
section in $f_{!}\mathcal{F}$.
|
||||
|
||||
Let $(U_i)_{i \in I}$ be a family of open sets in $Y$ and $s_i \in (f_{!} \mathcal{F})(U_i)$
|
||||
sections. Thus $s_i \in \mathcal{F}(f^{-1}(U_i))$ such that $\supp{s_i} \xrightarrow{f} U_i$
|
||||
is proper.
|
||||
Gluing yields a unique section $s \in \mathcal{F}(f^{-1}(U))$. We need
|
||||
to check that
|
||||
\[
|
||||
\supp{s} = \bigcup_{i \in I} \supp{s_i} \xlongrightarrow{f} \bigcup_{i \in I} U_i
|
||||
\] is proper. For this note that
|
||||
$\left(f|_{\supp{s}}\right)^{-1}(U_i) = f^{-1}(U_i) \cap \supp{s} = \supp{s_i}$ and
|
||||
being proper is local on the target.
|
||||
\end{proof}
|
||||
|
||||
\color{black}
|
||||
|
||||
\begin{bem}[Lower shriek is left exact]
|
||||
Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}''$ be an exact sequence
|
||||
in $\sh{X}$ and $f\colon X \to Y$ continuous. Then
|
||||
\[
|
||||
0 \to f_{!} \mathcal{F}' \to f_{!}\mathcal{F} \to f_{!}\mathcal{F}''
|
||||
\] is exact.
|
||||
\end{bem}
|
||||
|
||||
\color{gray}
|
||||
|
||||
\begin{proof}
|
||||
We have the following commutative diagram
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & f_{!} \mathcal{F}' \arrow{r} \arrow[hookrightarrow]{d}
|
||||
& f_{!} \mathcal{F} \arrow{r} \arrow[hookrightarrow]{d}
|
||||
& f_{!} \mathcal{F}'' \arrow[hookrightarrow]{d} \\
|
||||
0 \arrow{r} & f_{*} \mathcal{F}' \arrow{r}
|
||||
& f_{*} \mathcal{F} \arrow{r}
|
||||
& f_{*} \mathcal{F}''
|
||||
\end{tikzcd}
|
||||
,\] where the second row is exact. Thus the claim follows.
|
||||
\end{proof}
|
||||
|
||||
\color{black}
|
||||
|
||||
\begin{bem}[Lower shriek and compact support]
|
||||
Let $f\colon X \to \{ *\} $ be the unique continuous map from $X$ to the one point space
|
||||
and $\mathcal{F} \in \sh{X}$.
|
||||
Then
|
||||
\[
|
||||
(f_{!}\mathcal{F})(\{*\}) =
|
||||
\{ s \in \mathcal{F}(X)\colon \supp{s} \to \{ *\} \text{ proper}\}
|
||||
= \{ s \in \mathcal{F}(X)\colon \supp{s} \text{ compact}\}
|
||||
.\] Denote this by $\Gamma_c(X, \mathcal{F})$.
|
||||
\end{bem}
|
||||
|
||||
\section{Derivative of lower shriek}
|
||||
|
||||
The goal of this and the following talk is to prove the following theorem
|
||||
|
||||
\begin{theorem}[Verdier duality]
|
||||
If $X, Y$ are locally compact topological spaces of finite dimension,
|
||||
then $\mathrm{R}f_{!}$ admits a right adjoint
|
||||
$f^{!}\colon \mathrm{D}^{+}(Y) \to \mathrm{D}(X)$.
|
||||
\end{theorem}
|
||||
|
||||
To calculate the derivative of $f_{!}$, we need to introduce an adapted class of sheaves.
|
||||
|
||||
\begin{definition}
|
||||
Let $X$ be space, $\mathcal{F} \in \sh{X}$ and $Z \subseteq X$ a subset. Then
|
||||
define
|
||||
\[
|
||||
\mathcal{F}(Z) = \Gamma(Z, \mathcal{F}) = \Gamma(Z, \mathcal{F}|_{Z})
|
||||
\] where $\mathcal{F}|_{Z} = i^{*}\mathcal{F}$ for $i\colon Z \to X$ the canonical inclusion.
|
||||
\end{definition}
|
||||
|
||||
\begin{bem}[Lemma 1.4 in \cite{mustata}]
|
||||
If $\mathcal{F} \in \sh{X}$, $Z_1, Z_2 \subseteq X$ are closed
|
||||
and $t_1 \in \mathcal{F}(Z_1)$, $t_2 \in \mathcal{F}(Z_2)$ are given such that
|
||||
$t_1|_{Z_1 \cap Z_2} = t_2|_{Z_1 \cap Z_2}$, then
|
||||
there exists a unique section $t \in \mathcal{F}(Z_1 \cup Z_2)$ such that
|
||||
$t|_{Z_1} = t_1$ and $t|_{Z_2} = t_2$.
|
||||
\end{bem}
|
||||
|
||||
\color{gray}
|
||||
|
||||
\begin{bem}
|
||||
If $Z \subseteq X$ is a subset and $i\colon Z \to X$ the canonical inclusion, then
|
||||
\[
|
||||
\mathcal{F}(Z)
|
||||
=
|
||||
\left\{ (s_i, U_i)_{i \in I} \colon U_i \subseteq X \text{ open with } Z \subseteq \bigcup_{i \in I} U_i,
|
||||
s_i \in \mathcal{F}(U_i) \text{ with } (s_i)_z = (s_{j})_z \forall i, j \in I, z \in Z \cap U_i \cap U_j\right\} / \sim
|
||||
.\]
|
||||
where $(U_i, s_i)_{i \in I} \sim (V_j, t_j)_{j \in J}$
|
||||
if and only if $(s_i)_z = (t_j)_z$ for all $i \in I$, $j \in J$ and $z \in U_i \cap V_j \cap Z$.
|
||||
|
||||
For every open neighbourhood $U$ of $Z$, we have a restriction map
|
||||
\[
|
||||
\mathcal{F}(U) \to \mathcal{F}(Z), s \mapsto s|_Z \coloneqq [(s, U)]
|
||||
.\] This induces a map
|
||||
\[
|
||||
\colim{Z \subseteq U} \mathcal{F}(U)
|
||||
\to \mathcal{F}(Z)
|
||||
.\]
|
||||
\end{bem}
|
||||
|
||||
\begin{lemma}
|
||||
Let $X$ be a space and $\mathcal{F} \in \sh{X}$.
|
||||
If $Z \subseteq X$ is compact, the natural map
|
||||
\[
|
||||
\colim{Z \subseteq U} \mathcal{F}(U) \longrightarrow \mathcal{F}(Z)
|
||||
\] is an isomorphism.
|
||||
\end{lemma}
|
||||
|
||||
\begin{proof}
|
||||
Injectivity: Let $s \in \mathcal{F}(U)$ such that $s|_Z = 0$. Thus for all $z \in Z$,
|
||||
$s_z = 0$ and
|
||||
there exists an open neighbourhood
|
||||
$z \in U_z \subseteq U$ such that $s|_{U_z} = 0$. Thus $s|_{\bigcup U_z } = 0$. Since
|
||||
$Z \subseteq \bigcup_{z \in Z} U_z$, $s$ is zero in the colimit.
|
||||
|
||||
Surjectivity: Take $(s_i, U_i)_{i \in I} \in \mathcal{F}(Z)$. Thus
|
||||
$Z \subseteq \bigcup_{i \in I} U_i$ and by local compactness, for every $z \in Z$, there
|
||||
exists a compact neighbourhood $z \in K_z$ such that $K_z \subseteq U_{i_z}$ for
|
||||
some $i_z \in I$. Since $Z$ is compact, finitely many suffice, so we may assume
|
||||
$Z \subseteq \bigcup_{i=1}^{n} K_i$ and $K_i \subseteq U_i \subseteq X$.
|
||||
We now want to define a section on a neighbourhood of $Z$ that locally agrees with the $s_i$.
|
||||
|
||||
By induction, we may assume $n = 2$. By definition, $(s_1)_z = (s_2)_z$ for all $z \in Z \cap U_1 \cap U_2$,
|
||||
in particular $s_1|_{U_1 \cap U_2}$ and $s_2|_{U_1 \cap U_2}$ have the same restriction
|
||||
to $K_1 \cap K_2$. By the injectivity of the restriction map,
|
||||
there exists an open neighbourhood $K_1 \cap K_2 \subseteq V \subseteq U_1 \cap U_2$, such that
|
||||
$s_1|_V = s_2|_V$. Since $K_j \setminus V$ is closed in the compact $K_j$, for $j=1,2$
|
||||
the subset $K_j \setminus V$ is compact. Since $X$ is Hausdorff, there
|
||||
exist open neighbourhoods $K_j \setminus V \subseteq U_j' \subseteq U_j$ such that
|
||||
$U_1' \cap U_2' = \emptyset$. Now $s_1|_{U_1'}$, $s_2|_{U_2'}$ and
|
||||
$s_1|_V = s_2|_V$ glue to a section $w$ on $U_1' \cup U_2' \cup V \supseteq K_1 \cup K_2 \supseteq Z$
|
||||
such that $w|_Z = [(s_i, U_i)_{i \in I}]$.
|
||||
\end{proof}
|
||||
|
||||
\color{black}
|
||||
|
||||
\begin{definition}
|
||||
A sheaf $\mathcal{F} \in \sh{X}$ is \emph{soft} if
|
||||
$\mathcal{F}(X) \to \mathcal{F}(Z)$ is surjective whenever $Z \subseteq X$ is compact.
|
||||
\end{definition}
|
||||
|
||||
\begin{bem}
|
||||
In \cite{kashiwara} our notion of softness is called \emph{c-soft}.
|
||||
For $\sigma$-compact spaces the notions agree according to Exercise II.6 in \cite{kashiwara}.
|
||||
\end{bem}
|
||||
|
||||
\begin{bem}[Flasque sheaves are soft]
|
||||
Recall that a sheaf $\mathcal{F} \in \sh{X}$ is called \emph{flasque}, if
|
||||
for every open set $U \subseteq X$, the restriction map
|
||||
$\mathcal{F}(X) \to \mathcal{F}(U)$ is surjective. For $Z \subseteq X$ compact,
|
||||
we have a commutative diagram:
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\mathcal{F}(X) \arrow{rr} \arrow[twoheadrightarrow]{dr} & & \mathcal{F}(Z) \\
|
||||
& \colim{Z \subseteq U} \mathcal{F}(U) \arrow{ur}{\simeq} &
|
||||
\end{tikzcd}
|
||||
.\] Thus $\mathcal{F}$ is soft.
|
||||
\end{bem}
|
||||
|
||||
\begin{bem}[Prop. 2.5.6 in \cite{kashiwara}]
|
||||
Let $\mathcal{F} \in \sh{X}$. Then $\mathcal{F}$ is soft if and only if for
|
||||
any closed subset $Z \subseteq X$, the restriction
|
||||
$\Gamma_c(X, \mathcal{F}) \to \Gamma_c(Z, \mathcal{F}|_{Z})$
|
||||
is surjective.
|
||||
\end{bem}
|
||||
|
||||
\color{gray}
|
||||
|
||||
\begin{proof}
|
||||
If $K \subseteq X$ is compact, $\Gamma(K, F) = \Gamma_c(K, F|_K)$,
|
||||
so the condition is sufficient. Conversely
|
||||
assume $\mathcal{F}$ is soft and let $s \in \Gamma_c(Z, \mathcal{F}|_Z)$ with
|
||||
compact support $K$. Let $U$ be a relatively compact open neighbourhood of $K$ in $X$.
|
||||
Define $\tilde{s} \in \Gamma(\partial U \cup (Z \cap \overline{U}), \mathcal{F})$
|
||||
by setting $\tilde{s}_{Z \cap \overline{U}} = s$
|
||||
and $\tilde{s}|_{\partial U} = 0$. By softness, this extends to a global section
|
||||
$t \in \Gamma(X, \mathcal{F})$. Since $t = 0$ on a neighbourhood of $\partial U$,
|
||||
we may assume $t$ is supported by $\overline{U}$.
|
||||
\end{proof}
|
||||
|
||||
\color{black}
|
||||
|
||||
\begin{bsp}
|
||||
Let $M$ be a smooth manifold and let $f \in \mathcal{C}^{\infty}(K)$ be a
|
||||
section over a compact set $K$, i.e. a smooth function defined
|
||||
on some neighbourhood $U$ of $K$. Thus by using a partition of unity,
|
||||
we can extend $f$ to a global smooth function $\tilde{f} \in \mathcal{C}^{\infty}(M)$
|
||||
such that $\tilde{f}|_{K} = f$. In other words, the
|
||||
sheaf $\mathcal{C}^{\infty}$ is soft.
|
||||
|
||||
In a similar fashion we see that the sheaf of sections of a smooth vector bundle
|
||||
on $M$ is soft.
|
||||
\end{bsp}
|
||||
|
||||
\color{gray}
|
||||
|
||||
\begin{bsp}
|
||||
If $\mathcal{A}$ is a soft sheaf of rings and $\mathcal{F}$ is a sheaf of $\mathcal{A}$-modules,
|
||||
then $\mathcal{F}$ is soft. Indeed, let $s \in \mathcal{F}(K)$ be a section
|
||||
over a compact set $K \subseteq X$, i.e. a section on some open neighbourhood of $K$. By
|
||||
softness we can extend the section $1 \in \mathcal{A}(K)$ to a compactly supported global section
|
||||
$i \in \mathcal{A}(X)$ with support in $U$. Thus
|
||||
$si$ extends to a global section of $\mathcal{F}$.
|
||||
\end{bsp}
|
||||
|
||||
\begin{satz}
|
||||
Let $X$ be a space.
|
||||
If $\mathcal{F} \in \sh{X}$ is soft, $K \subseteq X$ is compact and $K \subseteq U$ is an open neighbourhood,
|
||||
any section over $K$ can be extended to a global section with compact support contained in $U$.
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
Let $s \in \mathcal{F}(K)$.
|
||||
By local compactness, there exists a compact neighbourhood $L$ of $K$ with $L \subseteq U$. Then
|
||||
$K \cap \partial L = \emptyset$. Consider the section on $K \cup \partial L$ given by
|
||||
$s$ on $K$ and zero on $\partial L$. Since $\mathcal{F}$ is soft, this can be extended
|
||||
to a global section, and a fortiori to a section $t$ over $L$. Now
|
||||
the sections given by $t$ on $L$ and $0$ on $\overline{X \setminus L}$ glue to a compactly
|
||||
supported extension of $s$. Since $L \subseteq U$, its support is contained in $U$.
|
||||
\end{proof}
|
||||
|
||||
\color{black}
|
||||
|
||||
\subsection{Compactly supported cohomology}
|
||||
|
||||
Let $X$ be a space.
|
||||
|
||||
%\begin{definition}
|
||||
% Let $U \subseteq X$ be open and $\mathcal{F} \in \sh{X}$. We define
|
||||
% $\Gamma_c(U, \mathcal{F})$ as the subgroup of $\Gamma(U, \mathcal{F})$ consisting of
|
||||
% sections with compact support.
|
||||
%\end{definition}
|
||||
%
|
||||
%\begin{bem}
|
||||
% If $s, t \in \Gamma(U, \mathcal{F})$ have compact support, so does $s + t$. Thus
|
||||
% $\Gamma_c(U, \mathcal{F})$ is indeed a subgroup of $\Gamma(U, \mathcal{F})$.
|
||||
%
|
||||
% Taking $U = X$, this defines a functor $\Gamma_c = \Gamma_c(X, \cdot)\colon \sh{X} \to \mathcal{A}b$
|
||||
%\end{bem}
|
||||
|
||||
\begin{theorem}[Base change]
|
||||
Let $f\colon X \to Y$ be a continuous map of spaces. For
|
||||
$\mathcal{F} \in \sh{X}$, there is a natural isomorphism
|
||||
\[
|
||||
(f_{!}\mathcal{F})_y \simeq \Gamma_c(f^{-1}(y), \mathcal{F}|_{f^{-1}(y)})
|
||||
\] for each $y \in Y$.
|
||||
\label{thm:base-change}
|
||||
\end{theorem}
|
||||
|
||||
\begin{proof}
|
||||
Denote by $X_y$ the fibre of $f$ over $y$ and by $\mathcal{F}$ the restriction to $X_y$.
|
||||
Let $y \in U \subseteq Y$ open. Then consider the natural map
|
||||
\begin{salign*}
|
||||
(f_{!}\mathcal{F})(U) &\longrightarrow \Gamma_c(X_y, \mathcal{F}_y) \\
|
||||
s &\longmapsto s|_{X_y}
|
||||
.\end{salign*}
|
||||
This is well-defined, since for any $s \in \mathcal{F}(f^{-1}(U))$ with
|
||||
$\supp{s} \xrightarrow{f} U$ proper, we have
|
||||
\[
|
||||
\supp{s|_{X_y}} = \supp{s} \cap X_y = \left( f|_{\supp{s}}^{U} \right)^{-1}(y)
|
||||
\] and the right hand side is compact. This map induces
|
||||
a natural map
|
||||
\[
|
||||
(f_{!}\mathcal{F})_y = \colim{y \in U \subseteq Y} (f_{!}\mathcal{F})(U)
|
||||
\longrightarrow \Gamma_c(X_y, \mathcal{F}_y)
|
||||
.\]
|
||||
|
||||
Injectivity: Let $s \in (f_{!}\mathcal{F})(U)$ such that $s|_{X_y} = 0$. Thus
|
||||
$s \in \mathcal{F}(f^{-1}(U))$ and $\supp{s} \xrightarrow{f} U$ is proper. Since
|
||||
$s|_{X_y} = 0$, $f^{-1}(y) \cap \supp{s} = X_y \cap \supp{s} = \emptyset$, in particular
|
||||
$y \not\in f(\supp{s})$. Let $y \in U'$ be the complement of $f(\supp{s})$ in $U$.
|
||||
Since $\supp{s} \xrightarrow{f} U$ is proper, $f(\supp{s})$ is closed in $U$, so
|
||||
$U'$ is open in $U$ and hence in $Y$. Moreover
|
||||
\[
|
||||
f^{-1}(U') \cap \supp{s}
|
||||
\subseteq f^{-1}(U') \cap f^{-1}(f(\supp{s}))
|
||||
= f^{-1}(U' \cap f(\supp{s}))
|
||||
= f^{-1}(\emptyset)
|
||||
= \emptyset
|
||||
.\]
|
||||
Hence $s|_{f^{-1}(U')} = 0$, so $s|_{U'} = 0$.
|
||||
|
||||
Surjectivity: Suppose first $\mathcal{F}$ is soft and let
|
||||
$s \in \Gamma_c(X_y, \mathcal{F}_y)$. Since $\mathcal{F}$ is soft, we may extend
|
||||
$s \in \mathcal{F}(X_y)$ to a compactly supported $s \in \mathcal{F}(X) = (f_{*}\mathcal{F})(Y)$.
|
||||
Since $Y$ is Hausdorff, every compact $K \subseteq Y$ is closed and therefore its preimage
|
||||
under $f|_{\supp{s}}$ is closed in the compact $\supp{s}$, thus itself compact. Hence
|
||||
$f|_{\supp{s}}\colon \supp{s} \to Y$ is proper and $s \in (f_{!}\mathcal{F})(Y)$.
|
||||
|
||||
For arbitrary $\mathcal{F}$, there exists an exact sequence
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & \mathcal{F} \arrow{r}
|
||||
& \mathcal{I} \arrow{r}
|
||||
& \mathcal{J}
|
||||
\end{tikzcd}
|
||||
\] with $\mathcal{I}, \mathcal{J}$ soft (e.g. injective). The functors
|
||||
$(f_{!} \cdot )_y$ and $\Gamma_c(X_y, \cdot |_{X_y})$ are left exact, so we have a commuting diagram
|
||||
with exact rows:
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & (f_!\mathcal{F})_y \arrow{r} \arrow{d}
|
||||
& (f_!\mathcal{I})_y \arrow{r} \arrow{d}{\simeq}
|
||||
& (f_!\mathcal{J})_y \arrow{d}{\simeq} \\
|
||||
0 \arrow{r} & \Gamma_c(X_y, \mathcal{F}_y) \arrow{r}
|
||||
& \Gamma_c(X_y, \mathcal{I}_y) \arrow{r}
|
||||
& \Gamma_c(X_y, \mathcal{J}_y)
|
||||
\end{tikzcd}
|
||||
.\] The five-lemma yields the desired isomorphism.
|
||||
\end{proof}
|
||||
|
||||
\begin{satz}[Lower shriek is exact on soft]
|
||||
Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ be an exact sequence
|
||||
in $\sh{X}$ with $\mathcal{F}'$ soft. Then the sequence
|
||||
\[
|
||||
0 \to f_{!}\mathcal{F}' \to f_{!}\mathcal{F} \to f_{!}\mathcal{F}'' \to 0
|
||||
\] is exact.
|
||||
\label{satz:lower-shriek-exact-on-soft}
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
Since $f_{!}$ is left exact, we only need to show the surjectivity on the right, i.e.
|
||||
for every $y \in Y$ the surjectivity of $(f_{!}\mathcal{F})_y \to (f_{!}\mathcal{F}'')_y$.
|
||||
We have the following commutative diagram:
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\Gamma_c(f^{-1}(y), \mathcal{F}|_{f^{-1}(y)}) \arrow{r} \arrow{d} & \arrow{d}
|
||||
\Gamma_c(f^{-1}(y), \mathcal{F}''|_{f^{-1}(y)}) \\
|
||||
(f_!\mathcal{F})_y \arrow{r} & (f_!\mathcal{F}'')_y
|
||||
\end{tikzcd}
|
||||
.\] By \ref{thm:base-change}, the vertical arrows are isomorphisms. It suffices
|
||||
thus to show the surjectivity of
|
||||
$\Gamma_c(f^{-1}(y), \mathcal{F}_{f^{-1}(y)}) \to \Gamma_c(f^{-1}(y), \mathcal{F}''|_{f^{-1}(y)})$.
|
||||
Restriction to $f^{-1}(y)$ is exact, moreover it preserves softness. We thus reduced
|
||||
to showing that $\Gamma_c(X, \cdot)$ preserves surjections.
|
||||
|
||||
Suppose first that $X$ is compact and let $s \in \Gamma_c(X, \mathcal{F}'') = \Gamma(X, \mathcal{F}'')$.
|
||||
Since $\mathcal{F} \to \mathcal{F}'' \to 0$ is exact, there exist
|
||||
a covering $X = \bigcup_{i \in I} U_i$ and lifts $t_i \in \mathcal{F}(U_i)$
|
||||
of $s|_{U_i}$. By local compactness of $X$, we may assume, after a possible refinement, that each
|
||||
$U_i$ contains a compact set $V_i$ whose interiors still cover $X$. Since
|
||||
$X$ is compact, we may assume $I$ is finite. To piece together the $t_i$, we may assume, by induction,
|
||||
that $\#I = 2$.
|
||||
|
||||
Consider $t_1|_{U_1 \cap U_2} - t_2|_{U_1 \cap U_2}$. This is necessarily a section $e'$ of
|
||||
$\mathcal{F}'(U_1 \cap U_2)$ as it maps to zero in $\mathcal{F}''(U_1 \cap U_2)$. Restricting
|
||||
$e'$ to the compact $V_1 \cap V_2$ and extending it by softness, yields a global section $e$ of
|
||||
$\mathcal{F}'$. Now
|
||||
\[
|
||||
(t_2|_{V_2} + e|_{V_2})|_{V_1 \cap V_2} = t_2|_{V_1 \cap V_2} + e'|_{V_1 \cap V_2} = t_1|_{V_1 \cap V_2}
|
||||
.\] Thus $t_1|_{V_1}, t_2|_{V_2} + e|_{V_2}$ glue to a global section $t$ of $\mathcal{F}$
|
||||
with image $s$.
|
||||
|
||||
Now for general $X$: Let $s \in \mathcal{F}''(X)$ with compact support $Z$. By local compactness,
|
||||
there exists a compact neighbourhood $Z' \subseteq X$ of $Z$. Since
|
||||
pullback of sheaves is exact and restriction of soft sheaves to closed subsets preserves softness,
|
||||
applying the result to $Z'$,
|
||||
yields a section $t' \in \mathcal{F}(Z')$ lifting $s|_{Z'}$. The restriction
|
||||
$t'|_{\partial Z'}$ maps to $s|_{\partial Z'} = 0$, so $t'|_{\partial Z'} \in \mathcal{F}'(\partial Z')$.
|
||||
Since $\partial Z'$ is compact and $\mathcal{F}'$ is soft, $t'|_{\partial Z'}$
|
||||
extends to a global section $b$ of $\mathcal{F}'$. Thus
|
||||
\[
|
||||
(t' - b|_{Z'})|_{\partial Z'} = t'|_{\partial Z'} - t'|_{\partial Z'} = 0
|
||||
.\] So
|
||||
$t' - b|_{Z'}$ on $Z'$ and $0$ on $\overline{X \setminus Z'}$ glue to a global section
|
||||
$t$ of $\mathcal{F}$. Then $t|_{Z'} = t' - b|_{Z'}$ maps to $s|_{Z'}$ since
|
||||
$b \in \mathcal{F}'(X)$. Since $\supp{t}, \supp{s} \subseteq Z'$, $t$ is a compactly supported lift of $s$.
|
||||
\end{proof}
|
||||
|
||||
\begin{korollar}
|
||||
If $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ is an exact sequence
|
||||
in $\sh{X}$ and $\mathcal{F}', \mathcal{F}$ are soft, then
|
||||
$\mathcal{F}''$ is soft too.
|
||||
\label{kor:soft-2+3}
|
||||
\end{korollar}
|
||||
|
||||
\begin{proof}
|
||||
Let $Z \subseteq X$ be closed.
|
||||
Since restricting to a closed subset is exact and preserves softness,
|
||||
by \ref{satz:lower-shriek-exact-on-soft}
|
||||
$\Gamma_c(Z, \mathcal{F}) \to \Gamma_c(Z, \mathcal{F}'')$ is surjective.
|
||||
This yields a commutative
|
||||
diagram
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
\Gamma_c(X, \mathcal{F}) \arrow[twoheadrightarrow]{d} \arrow{r} & \Gamma_c(X, \mathcal{F}'')
|
||||
\arrow{d} \\
|
||||
\Gamma_c(Z, \mathcal{F}) \arrow[twoheadrightarrow]{r} & \Gamma_c(Z, \mathcal{F}'')
|
||||
\end{tikzcd}
|
||||
,\] where the left vertical arrow is surjective, since $\mathcal{F}$ is soft. Since
|
||||
the composition is surjective, $\Gamma_c(X, \mathcal{F}'') \to \Gamma_c(Z, \mathcal{F}'')$ is also
|
||||
surjective.
|
||||
\end{proof}
|
||||
|
||||
\section{Derived categories and functors}
|
||||
|
||||
We give a brief introduction to the derived category of an abelian category $\mathcal{A}$. Let
|
||||
$F\colon \mathcal{A} \to \mathcal{B}$ be a left exact functor and let $\mathcal{A}$ have enough
|
||||
injectives. Then the classical derived functors exist. To compute $\mathrm{R}^{i}F(X)$ for an
|
||||
object $X \in \mathcal{A}$, we choose an injective resolution
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
0 \arrow{r} & X \arrow{r} \arrow{d} & 0 \arrow{r} \arrow{d} & 0 \arrow{r} \arrow{d} & \cdots\\
|
||||
0 \arrow{r} & I_0 \arrow{r} & I_1 \arrow{r} & I_2 \arrow{r} & \cdots
|
||||
\end{tikzcd}
|
||||
\] i.e. a quasiisomorphism $X \to \com{I} $. Then
|
||||
$\mathrm{R}^{i}F(X) = H^{i} F(\com{I})$.
|
||||
|
||||
New idea: identify $X$ with its resolution, in other words, turn quasiisomorphisms into
|
||||
isomorphisms. First step in this direction: Consider the category $\mathcal{K}(\mathcal{A})$
|
||||
of complexes where arrows are homomorphisms of complexes up to homotopy. Still
|
||||
quasiisomorphisms are in general not isomorphisms, so need to do more:
|
||||
|
||||
Localise by the class of quasiisomorphisms. This is then called the derived category
|
||||
of $\mathcal{A}$:
|
||||
\[
|
||||
\mathcal{D}(\mathcal{A}) = \mathcal{K}(\mathcal{A})_{\mathcal{Q}is}
|
||||
.\]
|
||||
Exactly like in the situation for rings, not every functor $\mathcal{K}(A) \to \mathcal{K}(B)$
|
||||
descends to the derived category, since it needs to send quasiisomorphisms to quasiisomorphisms.
|
||||
If the functor is induced by an exact functor $\mathcal{A} \to \mathcal{B}$, this is the case. For
|
||||
an arbitrary $F\colon \mathcal{A} \to \mathcal{B}$, we can hope that a derived functor exists. This
|
||||
is defined by a universal property, that ensures that this derived functor is in a sense
|
||||
close to the original one.
|
||||
|
||||
For a left exact functor $F\colon \mathcal{A} \to \mathcal{B}$, there is the following result:
|
||||
|
||||
\begin{theorem}
|
||||
If there exists a full additive subcategory $\mathcal{L}$ in $\mathcal{A}$ that is \emph{adapted} to $F$, i.e.
|
||||
\begin{enumerate}[(i)]
|
||||
\item for any $X \in \mathcal{A}$ there exists
|
||||
$X' \in \mathcal{L}$ and an exact sequence
|
||||
$0 \to X \to X'$
|
||||
\item if $0 \to X' \to X \to X'' \to 0$ is exact sequence in $\mathcal{A}$ and
|
||||
$X'$, $X$ are in $\mathcal{L}$, then $X''$ is in $\mathcal{L}$
|
||||
\item if $0 \to X' \to X \to X'' \to 0$ is exact sequence in $\mathcal{A}$ and
|
||||
if $X', X, X''$ are in $\mathcal{L}$, then the sequence
|
||||
$0 \to F(X') \to F(X) \to F(X'') \to 0$ is exact.
|
||||
\end{enumerate}
|
||||
Then the derived functor
|
||||
$\mathrm{R}F\colon \mathcal{D}^{+}(\mathcal{A}) \to \mathcal{D}^{+}(\mathcal{B})$
|
||||
exists and for any $\com{I} \in \mathcal{K}^{+}(\mathcal{L})$ we have a
|
||||
natural isomorphism
|
||||
\[
|
||||
\mathrm{R} F (\com{I}) \simeq F(\com{I})
|
||||
.\]
|
||||
\end{theorem}
|
||||
|
||||
Since $\sh{X}$ has enough injectives and every injective sheaf is soft, by
|
||||
\ref{satz:lower-shriek-exact-on-soft} and \ref{kor:soft-2+3}, the
|
||||
class of soft sheaves is adapted to the functor $f_!$. Thus the derived functor
|
||||
|
||||
\[
|
||||
\mathrm{R} f_{!} \colon \mathcal{D}^{+}(X) \longrightarrow \mathcal{D}^{+}(Y)
|
||||
\] exists.
|
||||
|
||||
\begin{korollar}
|
||||
For $\com{\mathcal{F}} \in \mathcal{K}om^{+}(\sh{X})$, we have a natural isomorphism
|
||||
\[
|
||||
(\mathrm{R}f_{!} \com{\mathcal{F}})_y
|
||||
\simeq \mathrm{R} \Gamma_c(f^{-1}(y), \com{\mathcal{F}}|_{f^{-1}(y)})
|
||||
\] in $\mathcal{D}(X)$.
|
||||
\end{korollar}
|
||||
|
||||
\begin{proof}
|
||||
Let $\com{\mathcal{F}} \to \com{\mathcal{I}}$ be an injective resolution. Then
|
||||
\begin{salign*}
|
||||
(\mathrm{R}f_{!} \com{\mathcal{F}})_y
|
||||
&\simeq (\mathrm{R}f_{!} \com{\mathcal{I}})_y \\
|
||||
&\simeq (f_{!} \com{\mathcal{I}})_y \\
|
||||
&\simeq \Gamma_c(f^{-1}(y), \com{\mathcal{I}}|_{f^{-1}(y)}) \\
|
||||
&\simeq \mathrm{R}\Gamma_c(f^{-1}(y), \com{\mathcal{I}}|_{f^{-1}(y)}) \\
|
||||
&\simeq \mathrm{R}\Gamma_c(f^{-1}(y), \com{\mathcal{F}}|_{f^{-1}(y)})
|
||||
.\end{salign*}
|
||||
\end{proof}
|
||||
|
||||
%\begin{korollar}
|
||||
% Soft sheaves are $\Gamma_c$-acyclic.
|
||||
% \label{kor:soft-gamma_c-acyclic}
|
||||
%\end{korollar}
|
||||
%
|
||||
%\begin{proof}
|
||||
% Let $\mathcal{F} \in \sh{X}$ be soft and
|
||||
% embed $\mathcal{F}$ in an injective sheaf $\mathcal{I}$. This yields an exact sequence
|
||||
% \[
|
||||
% \begin{tikzcd}
|
||||
% 0 \arrow{r} & \mathcal{F} \arrow{r}
|
||||
% & \mathcal{I} \arrow{r}
|
||||
% & \mathcal{G} \arrow{r}
|
||||
% & 0
|
||||
% \end{tikzcd}
|
||||
% .\]
|
||||
% Since $\mathcal{I}$ is injective, in particular flasque, hence soft,
|
||||
% by \ref{kor:soft-2+3}, $\mathcal{G}$ is soft.
|
||||
% We proceed by induction. For $i = 1$ consider the exact sequence
|
||||
% \[
|
||||
% \begin{tikzcd}
|
||||
% 0 \arrow{r} & \Gamma_c(X, \mathcal{F}) \arrow{r}
|
||||
% & \Gamma_c(X, \mathcal{I}) \arrow{r}
|
||||
% & \Gamma_c(X, \mathcal{G}) \arrow{r}
|
||||
% & H_c^{1}(X, \mathcal{F}) \arrow{r}
|
||||
% & \underbrace{H_c^{1}(X, \mathcal{I})}_{= 0}
|
||||
% \end{tikzcd}
|
||||
% .\]
|
||||
% Since $\mathcal{F}$ is soft, $\Gamma_c(X, \mathcal{I}) \to \Gamma_c(X, \mathcal{G})$ is
|
||||
% surjective. By the exactness of the sequence, $H_c^{1}(X, \mathcal{F})$ vanishes.
|
||||
% Now assume $H_c^{i}(X, \mathcal{F}) = 0$ for any soft sheaf $\mathcal{F}$. Then the exact sequence
|
||||
% \[
|
||||
% \begin{tikzcd}
|
||||
% \underbrace{H_c^{i}(X, \mathcal{I})}_{= 0} \arrow{r} & H_c^{i}(X, \mathcal{G}) \arrow{r}
|
||||
% & H_c^{i+1}(X, \mathcal{F}) \arrow{r}
|
||||
% & \underbrace{H_c^{i+1}(X, \mathcal{I})}_{= 0}
|
||||
% \end{tikzcd}
|
||||
% \] yields an isomorphism $H_c^{i}(X, \mathcal{G}) \simeq H_c^{i+1}(X, \mathcal{F})$ and
|
||||
% since $\mathcal{G}$ is soft, the left hand side is zero by induction hypothesis.
|
||||
%\end{proof}
|
||||
|
||||
%\begin{satz}
|
||||
% Soft sheaves are $f_!$-acyclic. In particular, if
|
||||
% $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ is an exact sequence in $\sh{X}$
|
||||
% and $\mathcal{F}'$ is soft, then the sequence
|
||||
% $0 \to f_!\mathcal{F}' \to f_!\mathcal{F} \to f_!\mathcal{F}'' \to 0$ is exact.
|
||||
%\end{satz}
|
||||
%
|
||||
%\begin{proof}
|
||||
% Let $i > 0$ and $\mathcal{F} \in \sh{X}$ be soft. Then for $y \in Y$
|
||||
% \begin{salign*}
|
||||
% (R^{i}f_!\mathcal{F})_y
|
||||
% \stackrel{\ref{thm:base-change}}{\simeq} H_c^{i}(f^{-1}(y), \mathcal{F}|_{f^{-1}(y)})
|
||||
% \; \stackrel{\ref{kor:soft-gamma_c-acyclic}}{=} \; 0
|
||||
% ,\end{salign*}
|
||||
% since the restriction of a soft sheaf to a closed subset is soft.
|
||||
%\end{proof}
|
||||
|
||||
\color{gray}
|
||||
|
||||
\begin{bsp}
|
||||
Let $U \subseteq X$ be open and $j\colon U \to X$ the inclusion map. By looking at stalks,
|
||||
one finds that $j_!\mathcal{F}$ for $\mathcal{F} \in \sh{U}$ is just extension by zero.
|
||||
\end{bsp}
|
||||
|
||||
\begin{satz}[Lower shriek preserves softness]
|
||||
If $f\colon X \to Y$ is continuous and $\mathcal{F} \in \sh{X}$ is soft, then
|
||||
$f_! \mathcal{F}$ is soft too.
|
||||
\end{satz}
|
||||
|
||||
\begin{proof}
|
||||
Let $Z \subseteq Y$ be compact and
|
||||
$s \in (f_!\mathcal{F})(Z) \simeq \colim{Z \subseteq U \subseteq Y} (f_!\mathcal{F})(U)$. Then
|
||||
there exists an open neighbourhood $U$ of $Z$ and an extension
|
||||
$\tilde{s} \in (f_!\mathcal{F})(U) \subseteq \mathcal{F}(f^{-1}(U))$ with
|
||||
$\supp{\tilde{s}} \xrightarrow{f} U$ proper. Since $Y$ is locally compact, there exists
|
||||
a compact neighbourhood $L \subseteq U$ of $Z$. Restricting $\tilde{s}$ to the compact
|
||||
$K \coloneqq \left(f|_{\supp{\tilde{s}}}\right)^{-1}(L) \subseteq \supp{\tilde{s}}$
|
||||
and extending by softness of $\mathcal{F}$, yields a compactly supported global section
|
||||
$t \in \mathcal{F}(X) = (f_{*}\mathcal{F})(Y)$ such that $t|_Z = s$. Since
|
||||
$\supp{t}$ is compact and $Y$ is Hausdorff, $\supp{t} \xrightarrow{f} Y$ is proper.
|
||||
\end{proof}
|
||||
|
||||
\begin{korollar}[Leray spectral sequence]
|
||||
Given continuous maps $f\colon X \to Y$, $g\colon Y \to Z$ of spaces,
|
||||
there is a natural isomorphism
|
||||
$\mathrm{R}(g \circ f)_{!} \simeq \mathrm{R}g_{!} \circ \mathrm{R}f_{!}$.
|
||||
\end{korollar}
|
||||
|
||||
\begin{proof}
|
||||
Since soft sheaves are $f_{!}$ (and $g_!$) acyclic and $f_{!}$ maps
|
||||
soft sheaves to soft sheaves, the result follows from
|
||||
Proposition 5.4 in \cite{hartshorne}.
|
||||
\end{proof}
|
||||
|
||||
\color{black}
|
||||
|
||||
\section{Other functors on abelian sheaves}
|
||||
|
||||
\begin{tabular}{l|l|l|l}
|
||||
Functor & Exactness & Derivative & Adapted class \\ \hline
|
||||
$f^* \colon \sh{Y} \to \sh{X}$
|
||||
& exact
|
||||
& $f^{*}\colon \mathcal{D}(Y) \to \mathcal{D}(X)$ \\
|
||||
$f_* \colon \sh{X} \to \sh{Y}$
|
||||
& left exact
|
||||
& $\mathrm{R} f_{*}\colon \mathcal{D}(X) \to \mathcal{D}(Y)$
|
||||
& K-limp complexes \\
|
||||
$\cdot \otimes \mathcal{F} \colon \sh{X} \to \sh{X}$
|
||||
& right exact
|
||||
& $\cdot \otimes^{L} \com{\mathcal{F}} \colon \mathcal{D}(X) \to \mathcal{D}(X)$
|
||||
& K-flat complexes \\
|
||||
$\underline{\operatorname{Hom}}(\mathcal{F}, \cdot)\colon \sh{X} \to \sh{X}$
|
||||
& left exact
|
||||
& $\mathrm{R}\com{\underline{\operatorname{Hom}}}(\com{\mathcal{F}}, \cdot)\colon \mathcal{D}(X) \to \mathcal{D}(X)$
|
||||
& K-injective complexes \\
|
||||
$f_!\colon \sh{X} \to \sh{Y}$
|
||||
& left exact
|
||||
& $\mathrm{R}f_{!}\colon \mathcal{D}^{+}(X) \to \mathcal{D}^{+}(Y)$
|
||||
& soft sheaves \\
|
||||
&
|
||||
& $f^{!}\colon \mathcal{D}^{+}(Y) \to \mathcal{D}(X)$
|
||||
\end{tabular}
|
||||
|
||||
\noindent The internal $\mathrm{Hom}$ functor is for $\mathcal{F}, \mathcal{G} \in \sh{X}$
|
||||
given by the formula
|
||||
\[
|
||||
\underline{\operatorname{Hom}}(\mathcal{F}, \mathcal{G})(U)
|
||||
= \operatorname{Hom}_{\sh{U}}(\mathcal{F}|_U, \mathcal{G}|_U)
|
||||
\] for every $U \subseteq X$ open and the (internal) tensor product by the sheafification
|
||||
of the presheaf
|
||||
\[
|
||||
U \mapsto \mathcal{F}(U) \otimes \mathcal{G}(U)
|
||||
.\]
|
||||
These functors satisfy the following adjunction results
|
||||
|
||||
\[
|
||||
f^{*} \dashv \mathrm{R}f_{*}
|
||||
\] and
|
||||
\[
|
||||
\cdot \otimes^{L} \com{\mathcal{F}} \dashv
|
||||
\mathrm{R}\underline{\operatorname{Hom}}(\com{\mathcal{F}}, \cdot )
|
||||
.\]
|
||||
|
||||
\bibliographystyle{alpha}
|
||||
\bibliography{refs}
|
||||
|
||||
\end{document}
|
||||
Submodule
+1
Submodule ws2022/rav/lecture added at 2f326caf56
Submodule
+1
Submodule ws2023/groupschemes-lecture added at ff678c3b5c
Reference in New Issue
Block a user