\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}