From 4c32de62c4f59254cc53c3ea2649bbd8c34c269b Mon Sep 17 00:00:00 2001 From: Christian Merten Date: Thu, 27 Apr 2023 12:05:43 +0200 Subject: [PATCH] 6ff: add talk notes --- notes.cls | 214 ++++++++++++ sose2023/6ff/refs.bib | 57 ++++ sose2023/6ff/vortrag.pdf | Bin 0 -> 222106 bytes sose2023/6ff/vortrag.tex | 565 +++++++++++++++++++++++++++++++ sose2023/6ff/vortrag2.pdf | Bin 0 -> 243826 bytes sose2023/6ff/vortrag2.tex | 682 ++++++++++++++++++++++++++++++++++++++ 6 files changed, 1518 insertions(+) create mode 100644 notes.cls create mode 100644 sose2023/6ff/refs.bib create mode 100644 sose2023/6ff/vortrag.pdf create mode 100644 sose2023/6ff/vortrag.tex create mode 100644 sose2023/6ff/vortrag2.pdf create mode 100644 sose2023/6ff/vortrag2.tex diff --git a/notes.cls b/notes.cls new file mode 100644 index 0000000..d6f3165 --- /dev/null +++ b/notes.cls @@ -0,0 +1,214 @@ +\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} + }, +} + +% 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 diff --git a/sose2023/6ff/refs.bib b/sose2023/6ff/refs.bib new file mode 100644 index 0000000..aecda60 --- /dev/null +++ b/sose2023/6ff/refs.bib @@ -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}}, +} diff --git a/sose2023/6ff/vortrag.pdf b/sose2023/6ff/vortrag.pdf new file mode 100644 index 0000000000000000000000000000000000000000..21b89ddeefbfeaf15f12affaab23f4211a402f4a GIT binary patch literal 222106 zcma&sLzFOFtRUL5ZQHhO+qP}nRbSb*ZQHhO+kF4))wA1!n?Y7)XLypGkSd6X(K6Dp zLXpldkE}y6au6^O*c(|v@$f*=%b41kyI2sgFfwrx{GWiL7qhf+F?AxK7qc;RF%>a2 zwl^__;^TvIc5yN_w1x86+|ki=-x@{rTdOnJEmua&^>viT+Z>h4Im#}LJ!EnF28fOa z35^~csr|X~HXmTx2bHjLbbd5Q0s;%p=<;#L<>T~!0mjv>ksm*|5)s?W{r>qro^UP@ zXKRY8Sok;@UyspbTa+@>+F1B-d3KYLCr?IT`dy=BK3bd8%TbaVtJCNC`gq>?aF@?z z-ty8@$h4F)e0`^YsnB2h@p;T%LPW(xkLm~~Us*z~WiF_v6tgE!?riblIC0Ub=1U$C 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+\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} diff --git a/sose2023/6ff/vortrag2.pdf b/sose2023/6ff/vortrag2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f37f8e3a1b94f12e261e1c6682ab80479c0d98ec GIT binary patch literal 243826 zcma&tQ*bU!)F$lMww>(QwrwXnwr$(CZQHi({r(>dLgCSp78Qp+kVIyKBvNy7V;pK&4kTtb4cd;O1W%_UTe-{jcxTTGYsS^={ zxQ(HUsi>*3y@@FdKR=AKi<7CLEsV$337(eo_Sie1KghSZ?9}UyG?%wU?wCT(EK57D 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100644 index 0000000..2097a1c --- /dev/null +++ b/sose2023/6ff/vortrag2.tex @@ -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}