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 0000000..21b89dd
Binary files /dev/null and b/sose2023/6ff/vortrag.pdf differ
diff --git a/sose2023/6ff/vortrag.tex b/sose2023/6ff/vortrag.tex
new file mode 100644
index 0000000..84543e2
--- /dev/null
+++ b/sose2023/6ff/vortrag.tex
@@ -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}
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+\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}