Sheaves, derived and perverse

Let $X,Y$ be topological spaces and $f:X\to Y$ a continuous map. We let $\Shv(X)$ be the category of sheaves on $X$, $D(\Shv(X))$ the derived category of sheaves on $X$, and $D_b(\Shv(X))$ the bounded variant. Recall that $D(\mathcal A)$ for an abelian category $\mathcal A$ is constructed first by taking $C(\mathcal A)$, the category of cochains of elements of $\mathcal A$, quotienting by chain homotopy, then quotienting by all acylic chains.

Remark: Let $\mathcal F\in \Shv(X)$. Recall:

• a section of $\mathcal F$ is an element of $\mathcal F(U)$ for some $U\subseteq X$,
• a germ of $\mathcal F$ at $x\in X$ is an equivalence class in $\{s\in \mathcal F(U)\ :\ U\owns x\}/\sim_x$,
• $s\sim_x t$ iff every neighborhood $W$ of $x$ in $U\cap V$ has $s|_W = t|_W$, for $s\in \mathcal F(U)$, $t\in \mathcal F(V)$,
• the support of the section $s\in\mathcal F(U)$ is $\supp(s) = \{x\in U\ :\ s \nsim_x 0\}$,
• the support of the sheaf $\mathcal F$ is $\supp(\mathcal F) = \{x\in X\ :\ \mathcal F_x\neq 0\}$.

Definition: The map $f$ induces functors between categories of sheaves, called
\[\begin{array}{r r c l}
\text{direct image} & f_*\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & (V\mapsto \mathcal F(f^{-1}(V))),\\[15pt]
\text{inverse image} & f^*\ :\ \Shv(Y) & \to & \Shv(X), \\
& (V\mapsto \mathcal G(V)) & \mapsto & \text{sh}\left(U\mapsto \text{colim}_{V\supseteq f(U)} \mathcal G(V)\right),\\[15pt]
\text{direct image with compact support} & f_!\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & \left(V\mapsto \left\{ s\in\mathcal F(f^{-1}(V))\ :\ f|_{\supp(s)} \text{ is proper}\right\}\right).
\end{array}\]

Above we used that $f:X\to Y$ is proper if $f^{-1}(K)\subseteq X$ is compact, for every $K\subseteq Y$ compact. Next, recall that a functor $\varphi:\mathcal A\to \mathcal B$ induces a functor $R\varphi:D(\mathcal A)\to D(\mathcal B)$, called the (first) derived functor of $\varphi$, given by $R\varphi(A^\bullet) = H^1(\varphi(A)^\bullet)$.

Remark: Each of the maps $f_*,f^*,f_!$ have their derived analogues $Rf_*, Rf^*,Rf_!$, respectively. For reasons unclear, $Rf_!$ has a right adjoint, denoted $Rf^!:D(\Shv(Y))\to D(\Shv(X))$. This is called the exceptional inverse image.

We are now ready to define perverse sheaves.

Definition: Let $A^\bullet \in D(\Shv(X))$. Then:

• the $i$th cohomology sheaf of $A^\bullet$ is $H^i(A^\bullet) = \ker(d^i)/\im(d^i)$,
• $A^\bullet$ is a constructible complex if $H^i(A^\bullet)$ is a constructible sheaf for all $i$,
• $A^\bullet$ is a perverse sheaf if $A^\bullet\in D_b(\Shv(X))$ is constructible and $\dim(\supp(H^{-i}(P))) \leqslant i$ for all $i\in \Z$ and for $P=A^\bullet$ and $P=(A^\bullet)^\vee = (A^\vee)^\bullet$ the dual complex of sheaves.

We finish off with an example.

Example: Let $X = \R$ be a stratified space, with $X_0=0$ the origin and $X_1 = \R\setminus 0$. Let $\mathcal F\in \Shv(X)$ be an $\R$-valued sheaf given by $\mathcal F(U) = \inf_{x\in U} |x|$, and define a chain complex $A^\bullet$ in the following way:
\[0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.\]
Note that for any $U\subseteq \R$, we have $H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset$ if $0\not\in U$, and $0$ otherwise. Hence $\supp(H^{-1}(A^\bullet)) = \R\setminus 0$, whose dimension is 1. Next, $H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0$, and so $\dim(\supp(H^0(A^\bullet))) = 0$. Note that $A^\bullet$ is self-dual and constructible, as the cohomology sheaves are locally constant. Hence $A^\bullet$ is a perverse sheaf.

References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")