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6ff: add talk notes

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Christian Merten před 2 roky
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Podepsáno: christian <christian@flavigny.de> ID GPG klíče: D953D69721B948B3
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@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}},
}

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\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}

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