\section{Intermediate extension (Allechar Serrano L\'opez)}

\[j_{!*}: \mathcal{D}^\heartsuit_U\rightarrow \mathcal{D}^\heartsuit \]

For $Y\in \mathcal{D}_U$, an extension of $Y$ to $\mathcal{D}$ is an $X\in\mathcal{D}$ and an isomorphism $j^*X\simeq Y$.

\begin{exmpl}
$j_!Y$ and $j_*Y$ are extensions of $Y$.
\end{exmpl}

\begin{defn}
Let $X\in\mathcal{D}^\heartsuit_U$. There is a canonical map $j_!X\rightarrow j_*X$. This induces a map ${ }^pj_!X\rightarrow {}{ }^pj_*X$. Then we define the intermediate extension

\[j_{!*}X=\text{im}({ }^pj_!X\rightarrow {}{ }^pj_!X)\]

where the image is taken in the abelian category $\mathcal{D}^\heartsuit$.

\end{defn}

\begin{lem}
$j_{!*}$ is the unique extension $\overline{X}$ of $X$ satisfying

\[\iota^*\overline{X}\in\mathcal{D}^{\leqslant -1}_Z  \text{ and }  \iota^{!}\overline{X}\in\mathcal{D}^{\geqslant 1}_Z\]
\end{lem}

\begin{proof}
We will show that $j_{!*}$ satisfies these properties. We have a surjection ${ }^pj_!X\twoheadrightarrow j_{!*}X$. Apply ${ }^p\iota^*$; this  is the left adjoint to ${ }^p\iota_*$, so it is right-exact, and we get a surjection

\[\underbrace{{ }^p\iota^{*}{}{ }^pj_!X}_{=0}\twoheadrightarrow {}{ }^p\iota^*j_{!*}X\]

so ${ }^p\iota^*\underbrace{j_{!*}X}_{\in \mathcal{D}^\heartsuit}=0$. Since $\iota^*$ is right-exact, we know that $\iota^*j_{!*}X\in \mathcal{D}^{\leqslant 0}_Z$; and we have just seen ${}^p H^0\iota^*j_{!*}X=0$, so we must have $\iota^*j_{!*}X\in\mathcal{D}^{\leqslant -1}_Z$.

For $\iota^!j_{!*}X$, argue similarly, applying ${ }^p\iota^!$ to the monomorphism $j_{!*}X\hookrightarrow {}^pj_*X$.

That shows existence. We leave uniqueness as an exercise.
\end{proof}
More generally:
\begin{prop}
For any $Y\in \mathcal{D}_U$ and $p\in \mathbb{Z}$, up to isomorphism, there exists a unique extension $X$ such that $\iota^*X\in \mathcal{D}^{\leqslant p-1}_Z$ and $\iota^!X\in \mathcal{D}^{\geqslant p+1}_Z$.
\end{prop}

The main idea is to define a new glued $t$-structure on $\mathcal{D}$ by glueing $(\mathcal{D}_U, 0)$ and $(\mathcal{D}^{\leqslant 0}_Z, \mathcal{D}^{\geqslant 0}_Z)$. The t-structure formalism then gives truncation functors  $^Z\tau^{\leqslant 0}$, $^Z\tau^{\geqslant 0}$ for this glued t-structure; and the extension promised in the Proposition is $X= {}^Z \tau^{\leq p-1}(j_* Y)$.


The following result gives a third characterization of $j_{!*}$:

\begin{lem}\label{LLemma}
Consider $X\in\mathcal{D}^\heartsuit_U$, then $j_{!*}$ is the unique perverse extension of $X$ having no non-trivial subobject or quotient in $\iota_*(\mathcal{D}^\heartsuit_Z)$.
\end{lem}

\begin{proof}
$j_{!*}X$ has no quotient of this form since ${ }^p\iota^*j_{!*}X=0$ (and ${ }^p\iota^*$ is right $t$-exact) and has no subextension because ${ }^p\iota^!j_{!*}X=0$ (and ${ }^p\iota^!$ is left $t$-exact).

Now we check that these two properties uniquely characterize a perverse extension:

Suppose $\overline{X}$, $j^*\overline{X}\simeq X$ is an extension. We have a surjection $\overline{X}\rightarrow { }^p\iota_*{ }^p\iota^*\overline{X}\rightarrow 0$. $\overline{X}$ has no nonzero quotient of this form, so ${ }^p\iota_*{ }^p\iota^*\overline{X}=0$, so by \ref{BLemma} we get $\underbrace{{ }^pj_!{ }^pj^*\overline{X}}_{={ }^pj_!X}\twoheadrightarrow \overline{X}$.
The dual argument shows $\overline{X}\hookrightarrow { }^pj_*X$, and the composite of these two maps to and from $\overline{X}$ is the canonical map, i.e., $\overline{X}=\text{im}({ }^pj_!X\rightarrow { }^pj_*X)$ .
\end{proof}
We can use what we've shown to classify the simple objects of $\mathcal{D}^{\heartsuit}$:
\begin{prop}\label{simples}
The simple objects of $\mathcal{D}^\heartsuit$ are those of the form ${ }^p\iota_*S_Z$ for $S_Z$ simple in $\mathcal{D}^\heartsuit_Z$ or $j_{!*}S_U$ for $S_U$ simple in $\mathcal{D}^\heartsuit_U$.
\end{prop}

\begin{proof}
Let $S\in \mathcal{D}^\heartsuit$ be simple. There are two cases:

\begin{enumerate}
\item[(i)] If ${ }^p\iota^*S\neq 0$, then $S\rightarrow { }^p\iota_*{ }^p\iota^*S$ (\ref{BLemma}) is a nonzero quotient of $S$. $S$ is simple, so $S\simeq { }^p\iota_*{ }^p\iota^*S$.
\item[(ii)] ${ }^p\iota^*S=0$. We may also assume ${ }^p\iota^!S=0$, else we would get a nonzero subobject of $X$ from ${ }^p\iota_*{ }^p\iota^!S$ (and again since $X$ is simple this inclusion would have to be an isomorphism).
\end{enumerate}

By \ref{LLemma}, we have that $S\simeq j_{!*}j^*S$, so we have to check $j^*S\in\mathcal{D}^\heartsuit_U$ is simple. Suppose not, so there exists a non-trivial quotient $j^*S\twoheadrightarrow Q\in\mathcal{D}^\heartsuit_U$. Then we have the diagram

\begin{center}
\begin{tikzcd}
{ }^pj_!j^*S \arrow[r, two heads] \arrow[d, two heads] &{ }^pj_!Q \arrow[d, two heads]\\
j_{!*}j^*S=S\arrow[r, dashed] &j_{!*}Q
\end{tikzcd}
\end{center}

Since the other three arrows are surjective, we can deduce $j_{!*}j^*S\twoheadrightarrow j_{!*}Q$ is surjective; since $S$ is simple, this is an isomorphism, hence $j^*S\rightarrow Q$ is an isomorphism.

To finish the proof, we need to check that $j_{!*}(\text{simple})=\text{simple}$. If not, there exists a proper quotient $\alpha:j_{!*}S_U\twoheadrightarrow Q$. Since $S_U$ is simple, $j^*(\alpha)$ is either zero or an isomorphism. If $j^*\alpha=0$, then $j^*Q=0$ so $Q\simeq { }^p\iota_*{ }^p\iota_*Q$ which contradicts \ref{LLemma}. So we may assume $j^*\alpha$ is an isomorphism, i.e., $j^*(\ker\alpha)=0$. (We have repeatedly used that $j^*$ is t-exact.) From the following short exact sequence

\[{ }^pj_!{ }^pj^*\ker\alpha \rightarrow \ker\alpha \rightarrow { }^p\iota_*{ }^p\iota^*\ker\alpha\rightarrow 0\]

we get that $\ker\alpha\simeq { }^p\iota_*{ }^p\iota^*\ker\alpha$, but $j_{!*}(\hspace{.3 cm})$ has no no-zero subobject in ${ }^p\iota_*(\mathcal{D}^\heartsuit_Z)$. Thus, $\ker\alpha=0$.
\end{proof}

\begin{prop}
$j_{!*}: \mathcal{D}^\heartsuit_U\rightarrow\mathcal{D}^\heartsuit$ is fully faithful.
\end{prop}

\begin{proof}
Consider $A, B\in\mathcal{D}^\heartsuit_U$. We have maps 

\[\Hom (A, B)\xrightarrow{j_{!*}}\Hom (j_{!*}A, j_{!*}B)\xrightarrow{j^*}\Hom (A, B),\]

and $j^*j_{!*}$ is the identity on $\Hom (A, B)$. To show $j_{!*}$ is bijective, we need to check that $j_{!*}j^*$ is the identity on $\Hom(j_{!*}A, j_{!*}B)$. Since $j_{!*}B\subset { }^pj_*B$, we get

\begin{align*}
\Hom(j_{!*}A, j_{!*}B)\subset &\Hom(j_{!*}A, { }^pj_*B)\\
&=\Hom({ }^pj^*j_{!*}A, B)\\
&=\Hom(A, B).
\end{align*}

Now, for any $\phi \in \Hom(j_{!*}A, j_{!*}B)$ to show $\phi=j_{!*}j^*\phi$, it suffices to show via this injective composite that $j^*\phi=j^*j_{!*}j^*\phi$. This holds because $j_{!*}j^*\simeq \text{id}$.
\end{proof}
\section{Examples. Poincar\'e Duality.}
Now, we return to algebraic geometry. Consider $X$ over $k$ separated of finite type for $k$ a perfect field. The basic example of a perverse sheaf is given $U$ smooth of dimension $d$ and $\mathcal{F}\in\text{Loc}(U)$, consider $X=\mathcal{F}[d]\in\text{Perv}(U)$. Now, use $j_{!*}$. 
\begin{defn}
For any $X$ of (pure) dimension $d$, let $U$ be a smooth open dense sub-scheme, and let $\mathcal{F}\in\text{Loc}(U)$, so $j_{!*}\mathcal{F}[d]\in\text{Perv}(X)$. We define $IC_X(\mathcal{F})=j_{!*}\mathcal{F}[d]\in\text{Perv}(X)$; in particular, the intersection cohomology complex of $X$ is by definition $IC_X=j_{!*}\mathbb{Q}_{l}[d]$.
\end{defn}

\begin{defn}
The intersection cohomology groups of $X$ with coefficients in $\mc{L} \in \mathrm{Loc}(U)$ (some smooth open dense $U \subset X$) are by definition $IH^n(X, \mathcal{L})=H^{n-d}(X, IC_X(\mathcal{L}))=H^{n-d}(R\Gamma(X, IC_X(\mathcal{L})))$. This $n-d$ normalization fixes things so that for $X$ smooth we have $IH^n=H^n$.
\end{defn}

\begin{exmpl}
Assume $U=X\smallsetminus \{x\} \arrowcircle X$ is smooth. Then $IC_X=\tau^{\leqslant -1}(Rj_*\mathbb{Q}_l[d])$. 
\begin{proof}
Recall that, in general, $IC_X$ is characterized as the unique extension of $\mathbb{Q}_l[d]\vert_U$ such that

\[\iota^*IC_X\in { }^pD^{\leqslant -1}(Z)\text{ and } \iota^!IC_X\in { }^pD^{\geqslant 1}(Z)\]

where $j: U\arrowcircle X$ and $\iota: Z=X\smallsetminus U\hookrightarrow X$

Since $i^*$ is exact for the standard t-structure, $i^* \tau^{\leq -1}(Rj_* \mathbb{Q}_l[d])= \tau^{\leq -1} i^* Rj_*\mathbb{Q}_l[d] \in \mathcal{D}^{\leq -1}_{\{x\}}= {}^p \mathcal{D}^{\leq -1}_{\{x\}}$, since $\dim \{x\}=0$. Similarly, to compute 

\[\iota^!(\tau^{\leqslant -1}Rj_*\mathbb{Q}_l[d])\]

we need to look at the distinguished triangle

\[\iota^!\tau^{\leqslant -1}(Rj_*\mathbb{Q}_l[d])\rightarrow \iota^!Rj_*\mathbb{Q}_l[d]\rightarrow \iota^!\tau^{\geqslant 0}(Rj_*\mathbb{Q}_l[d])\xrightarrow{+1}.\]

Since $i^! Rj_*=0$, we find

\[\underbrace{\underbrace{\underbrace{\iota^!}_{\text{left-exact}}\tau^{\geqslant 0}(Rj_*\mathbb{Q}_l[d])}_{\mathcal{D}^{\geqslant 0}(\{x\})}[-1]}_{\mathcal{D}^{\geqslant 1}(\{x\})= { }^pD^{\geqslant 1}(\{x\})}\xrightarrow{\sim}\iota^!\tau^{\leqslant -1}(Rj_*\mathbb{Q}_l[d])\],
so by uniqueness we find $IC_X \equiv \tau^{\leq -1}(Rj_* \mathbb{Q}_l[d])$.
\end{proof}
\end{exmpl}

\begin{rmk}
This formula generalizes to Deligne's successive truncation (coming from a smooth stratification of $X$) formula for $j_{!*}$
\end{rmk}

\begin{exmpl}
Continue in the setting of the last example: $j:U=X\smallsetminus \{x\}\arrowcircle X$ and $\iota:Z=\{x\}\hookrightarrow X$, with $U$ smooth. We aim to compute the intersection cohomology $IH^\bullet(X)=H^\bullet(X, IC_X[-d])$. Since $IC_X=\tau^{\leqslant -1}(Rj_*\mathbb{Q}_l[d])$, $IC_X[-d]=\tau^{\leqslant d-1}(Rj_*\mathbb{Q}_l)$ so we have the following distinguished triangle

\[IC_X[-d]\rightarrow Rj_*\mathbb{Q}_l\rightarrow \tau^{\geqslant d}Rj_*\mathbb{Q}_l\xrightarrow{+1}\]

Taking the long exact sequence in hypercohomology (i.e., apply $R\Gamma(X, \cdot)$ and then take the LES in cohomology), we obtain

\[\cdots\rightarrow IH^r(X)\rightarrow H^r(U)\rightarrow H^r(X, \tau^{\geqslant d}Rj_*\mathbb{Q}_l)\rightarrow IH^{r+1}(X)\rightarrow\cdots,\]
where we have rewritten the $H^r(U)$ term using the Leray spectral sequence for $U \to X \to \mathrm{Spec}(k)$.

The calculation splits into three cases, namely:

\begin{itemize}
\item[1)] $r<d$
\item[2)] $r=d$
\item[3)] $r>d$
\end{itemize}

We will expand on each of these cases.

\begin{itemize}
\item[1)] ($r < d$) We have $IH^r(X)\xrightarrow{\sim}H^r(X\smallsetminus\{x\})$. Indeed, there is a hypercohomology spectral sequence

\[E^{a, b}_2=H^a(X, H^b(K)) \implies H^{a+b}(X, K)\]

for $K\in D^b(X)$. Apply this to $K=\tau^{\geqslant d}Rj_*\mathbb{Q}_l$. For $b<d$, $H^b(\tau^{\geqslant d}Rj_*\mathbb{Q}_l)=0$. So for $r<d$, all relevant $E_2$ terms are already zero, and we find $H^r(X, \tau^{\geqslant d}Rj_*\mathbb{Q}_l)=0$.

\item[2) and 3)] ($r \geq d$) We have the distinguished triangles

\begin{center}
\begin{tikzcd}
\tau^{\leqslant 0}Rj_*\mathbb{Q}_l=\mathbb{Q}_l\arrow[r]\arrow[d] &Rj_*\mathbb{Q}_l\arrow{r}\arrow[d, equal] &\tau^{\geqslant 1}Rj_*\mathbb{Q}_l\arrow[r, "+1"]\arrow[d, dashed, "\exists"] &\cdots\\
\tau^{\leqslant d-1}Rj_*\mathbb{Q}_l=IC_X[-d]\arrow[r] &Rj_*\mathbb{Q}_l\arrow[r] &\tau^{\geqslant d}Rj_*\mathbb{Q}_l\arrow[r, "+1"] &\cdots
\end{tikzcd}
\end{center}

and we get 

\begin{center}
\begin{tikzcd}
\cdots\arrow[r] &H^r(X)\arrow[r] \arrow[d] &H^r(U)\arrow[r]\arrow[d, equal] &H^r(X, \tau^{\geqslant 1}Rj_*\mathbb{Q}_l)\arrow[r]\arrow[d] &H^{r+1}(X)\arrow[r]\arrow[d]&\cdots\\
\cdots \arrow[r] &IH^r(X)\arrow[r] &H^r(U)\arrow[r] &H^r(X, \tau^{\geqslant d}Rj_*\mathbb{Q}_l)\arrow[r] &IH^{r+1}(X)\arrow[r] &\cdots
\end{tikzcd}
\end{center}

By 1), for $r=d$ we have

\[0\rightarrow IH^d(X)\rightarrow H^d(U)\]

so we deduce $IH^d(X)\simeq \text{im}(H^d(X)\rightarrow H^d(X\smallsetminus x))$. 

For 3), we claim that $H^r(X)\simeq IH^r(X)$ for $r\geqslant d+1$. Indeed, the top triangle is isomorphic to the (adjunction) distinguished triangle

\[\mathbb{Q}_l\rightarrow Rj_*\mathbb{Q}_l\rightarrow \iota_*R\iota^!\mathbb{Q}_l[1]\xrightarrow{+1}\]

so the cohomology groups in the third column above are $H^r(X, \iota_*R\iota^!\mathbb{Q}_l[1])=H^r(\{x\}, R\iota^!\mathbb{Q}_l[1])$ and $H^r(X, \tau^{\geqslant d}\iota_*R\iota^!\mathbb{Q}_l[1])=H^r(\{x\}, \tau^{\geqslant d}R\iota^!\mathbb{Q}_l[1])$.

In the corresponding hypercohomology spectral sequences, only $E^{0, r}_2$ terms could be nonzero; but for $r \geq d$, $H^r(R\iota^!\mathbb{Q}_l[1])=H^r(\tau^{\geqslant d}R\iota^!\mathbb{Q}_l[1])$. A diagram chase then shows that for $r \geq d+1$, the maps $H^r(X)\xrightarrow{\sim} IH^r(X)$ are themselves isomorphisms.
\end{itemize}

\begin{exmpl}
Let $X$ be a pinched torus (one nodal singularity). Then by the above calculation,

\begin{equation*}
IH^r(X)=\begin{cases}
H^0(U)=\mathbb{Q}, &r=0\\
\text{im}(H^1(X)\rightarrow H^1(U))=0, &r=1\\
H^2(X)=\mathbb{Q}, &r=2
\end{cases}
\end{equation*}

In contrast, $H^1(X)=\mathbb{Q}$.
\end{exmpl}

\begin{exmpl}
Let $X$ be the union of two lines in $\mathbb{P}^2$. Topologically, $X=S^2\vee S^2$. Poincar\'e duality fails since we have

\begin{equation*}
H^r(X)=\begin{cases}
\mathbb{Q}, &r=0\\
0, &r=1\\
\mathbb{Q}\oplus\mathbb{Q}, &r=2
\end{cases}
\end{equation*}

In contrast,

\begin{equation*}
IH^r(X)=\begin{cases}
H^0(U)=\mathbb{Q}\oplus\mathbb{Q}, &r=0\\
0, &r=1\\
H^2(X)=\mathbb{Q}\oplus\mathbb{Q}, &r=2
\end{cases}
\end{equation*}

so at least at the level of Betti numbers, Poincar\'e duality is restored.

\end{exmpl}

\end{exmpl}

Indeed, one of the original motivations for Goresky and Macpherson to define intersection cohomology was to find a version of Poincar\'e duality for singular spaces. We'll now see that this works in general for intersection cohomology.
\begin{thm}
Let $X$ be any variety of dimension $d$. Then $IH^r(X)\xrightarrow{\sim}\underbrace{IH^{2d-r}_c(X)^\vee}_{H^{2d-r}(R\Gamma_c(X, IC_X[-d]))}$. More generally, for $j:U\arrowcircle X$,

\[\mathbb{D}_X\circ j_{!*}\simeq j_{!*}\circ\mathbb{D}_U\]

on $\text{Perv}(U)$.
\end{thm}

\begin{proof}
Let $K\in\text{Perv}(U)$. Set $L=\mathbb{D}_X\circ j_{!*}K$. Then $j^*L=j^*\mathbb{D}_Xj_{!*}K=\mathbb{D}_Uj^*j_{!*}K=\mathbb{D}_UK$ so $L$ extends $\mathbb{D}_UK$.

To show $L\cong j_{!*}\mathbb{D}_UK$ we have to check the support and co-support conditions on $\iota^*L$ and $\iota^!L$. We have 

\begin{align*}
\iota^*L&=\iota^*\mathbb{D}_Xj_{!*}K\\
&=\mathbb{D}_Z\underbrace{\iota^!j_{!*}K}_{\in {}^p D^{\geqslant 1}(Z)}\in {}^pD^{\leqslant -1}(Z)
\end{align*}

and likewise,

\begin{align*}
\iota^!L&=\iota^!\mathbb{D}_Xj_{!*}K\\
&=\mathbb{D}_Z\underbrace{\iota^*j_{!*}K}_{\in {}^pD^{\leqslant -1}}\in {}^pD^{\geqslant 1}(Z)
\end{align*}

For Poincar\'e duality, take $U$ smooth open dense in $X$; applying the above, and our previous calculation of $\mathbb{D}_U$ on lisse sheaves (for smooth $U$), we find $\mathbb{D}_X(IC_X)\simeq IC_X$ so $R\Gamma(X, IC_X)\simeq R\Gamma(X, \mathbb{D}_XIC_X)$. Letting $\pi \colon X \to \mathrm{Spec}(k)$ be the structure map, the relation $\mathbb{D}_{\mathrm{Spec}(k)} R \pi_! \equiv R \pi_* \mathbb{D}_X$ translates to an isomorphism $R\Gamma(X, \mathbb{D}_XIC_X)\simeq R\Gamma_c(X, IC_X)^\vee$. Taking cohomology, we find $H^n(X, IC_X)\simeq H^{-n}_c(X, IC_X)^\vee$. i.e., $IH^{d+n}(X)\simeq IH^{d-n}_c(X)^\vee$.
\end{proof}

\section{Statements of a few more basic facts about perverse sheaves in algebraic geometry}
We conclude with a few important facts about perverse sheaves that we won't have time to prove. See BBD for details.
\begin{itemize}
\item[1.] Let $X$ be a variety.
\begin{itemize}
\item[a)] Classification of simple objects in $\text{Perv}(X)$: all have the form $\iota_*j_{!*}\mathcal{L}[\dim Z]$ for a closed immersion $\iota:Z\hookrightarrow X$ and a local system $\mathcal{L}$ on a smooth dense open $W \subset Z$. (This is very easy to prove from Proposition \ref{simples}.)
\item[b)] $\text{Perv}(X)$ is artinian and noetherian (again, easy at this point).
\end{itemize}
\item[2.] Beilinson showed $\text{Perv}(X)\subset D^b_c(X)$ extends to an equivalence

\[D^b(\text{Perv}X)\xrightarrow{\sim}D^b_c(X).\]

This statement is not at all obvious; the key geometric point, which has many uses, is:

\item[3)] If $f:X\rightarrow Y$ is affine, then $Rf_*$ is $t$-right exact and $Rf_!$ is $t$-left exact.
\end{itemize}
The following is a substantial theorem. It is proven in BBD by reduction to the case of finite fields, where the argument rests on Deligne's relative version of the Weil conjectures (Weil II). Over the complex numbers there are also proofs (Saito, DeCataldo-Migliorini) via Hodge theory.
\begin{thm}[The Decomposition Theorem]
Let $f: X\rightarrow Y$ be a proper map of varieties over $k$. Then there exists a non-canonical isomorphism in $D^b_c(Y)$

\[Rf_*IC_X\simeq \oplus_{i\in\mathbb{Z}} {}^pH^i(Rf_*IC_X)[-i],\]

and for each $i$ there exists a canonical decomposition

\[{}^pH^i(Rf_*IC_X)=\oplus_{\alpha}IC_{\overline{Y}_{\alpha}}(\mathcal{L}_\alpha[\dim Y_\alpha])\]

where $Y= \sqcup_{\alpha} Y_\alpha$ is a stratification into smooth locally-closed subschemes, and each $\mathcal{L}_{\alpha}$ is a $\underbrace{\text{geometrically semi-simple}}_{\substack{\text{i.e., semi-simple as a }\\ \pi^{\acute{e}t}_1(Y_{\alpha, \overline{k}})\text{-representation}}}$ local system on $Y_{\alpha}$.
\end{thm}

To give a sense for what this theorem means, here is one of its classical precursors. If $f:X\rightarrow Y$ is a smooth projective (or proper) map of smooth varieties, then the Leray spectral sequence 
\[
E^{a, b}_2=H^a(Y, R^bf_*\mathbb{Q}_l) \implies H^{a+b}(X, \mathbb{Q}_l)
\]
degenerates at the $E_2$ page, and the local systems $R^bf_*\mathbb{Q}_l$ are geometrically semi-simple (Deligne). Even better, we have a splitting

\[Rf_*\mathbb{Q}_l\xrightarrow{\sim}\oplus_{i\in\mathbb{Z}}R^if_*\mathbb{Q}_l[-i]\]

\begin{exmpl}
Note that in general the Leray spectral sequence does not degenerate at $E_2$: consider any resolution of singularities $f:X\rightarrow Y$ where $H^n(Y)$ is not pure (in the sense of Hodge theory or the Frobenius action over finite fields) for some $n$.
\end{exmpl}

So one perspective on the decomposition theorem is that it generalizes Deligne's Leray degeneration theorem to arbitrary proper maps.

Time's up! But now we at least have the necessary foundations for geometric Satake.