\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
\[
\cdot \otimes^{L} \com{\mathcal{F}} \dashv
\mathrm{R}\underline{\operatorname{Hom}}(\com{\mathcal{F}}, \cdot )
.\] 

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