At the end one has an $\underline{E}^\times$-equivariant diagram with cartesian squares

$$
\begin{tikzcd}
BC(s^*T_{Z/X_S}) \arrow[r] \arrow[d, "\cong"] & \mathcal{M}_W^{sm} \arrow[r] \arrow[d, "g"] & \mathcal{M}_Z^{sm} \times \underline{E} \setminus \{0\} \arrow[d, "f"] \\
\text{zero section} \arrow[r, "\xrightarrow{\text{Id}_S \times \{0\}}"] & S \arrow[r, "\cong"] & S \times \underline{E} \arrow[r, "\xrightarrow{\text{Id}_E \setminus \{0\}}"] & S \times \underline{E} \setminus \{0\}
\end{tikzcd}
$$

**2.3. Construction of the deformation: 2nd step.** Up to replacing $W$ by a quasi-compact open nbd. of the section $X_S \times \mathbb{B}^1 \hookrightarrow W$ we can suppose we have an $\underline{O}_E \setminus \{0\}$-equivariant diagram

$$
\begin{tikzcd}
& & U \arrow[r] \arrow[d, "g"] & \mathcal{M}_W \arrow[r] \arrow[d, "f"] & \Delta \arrow[d, "\cong"] \\
\text{zero section} & S \arrow[r, "\cong"] & S \times_{\underline{O}_E} \Delta \arrow[r, "\cong"] & S \times_{\underline{O}_E} \{0\} & S \times_{\underline{O}_E} \{0\}
\end{tikzcd}
$$

with $M_W = \mathcal{M}_W^{sm}$ that is quasi-compact and $U$ an open nbd. of the zero section of $BC(s^*T_{Z/X_S})$. Pulling-back the situation to $\pi^{\mathbb{N}}\cup\{+\infty\} \subset \mathcal{O}_E$ we obtain a $\pi^{\mathbb{N}}$-equivariant diagram

$$
\begin{tikzcd}
& & U \arrow[r] \arrow[d, "g"] & \mathcal{M}_W \arrow[r] \arrow[d, "f"] & \Delta \arrow[d, "\cong"] \\
\text{zero section} & S \arrow[r, "\cong"] & S \times_{\pi^{\mathbb{N}}\cup\{+\infty\}} \Delta \arrow[r, "\cong"] & S \times_{\pi^{\mathbb{N}}} \{0\} & S \times_{\pi^{\mathbb{N}}} \{0\}
\end{tikzcd}
$$

**2.4. Proof of the Jacobian criterion.** Let

$$A = Rg' \mathbb{F}_{\ell} \in D_{\text{ét}}(M_W, \mathbb{F}_{\ell})$$

that is $\pi^{\mathbb{N}}$-equivariant. If $i: U \hookrightarrow \mathcal{M}_W^{sm}$, since $\mathbb{F}_{\ell}$ is $g$-ULA,

$$i^*A \in D_{\text{ét}}(U, \mathbb{F}_{\ell})$$

is étale locally on $S$ isomorphic to $\mathbb{F}_{\ell}[2d]$ with the trivial $\pi^{\mathbb{N}}$-equivariant structure, where $d = \text{deg}(s^*T_{Z/X_S})$. Since $S$ is strictly totally disconnected we can fix an $\underline{E}^\times$-equivariant isomorphism

$$\mathbb{F}_{\ell}[2d] \xrightarrow{\sim} i^*A.$$

Now, we have

$$\lim_{N \to \infty} H^{-2d} \left( \underbrace{g^{-1} \left( S \times \pi^{\mathbb{N} \cup \{+\infty\}} \right)}_{\mathcal{M}_W^{\ge N}}, A \right) \xrightarrow{\sim} H^{-2d}(U, i^*A).$$
We can thus find an integer $N \ge 0$ and a $\pi^N$-equivariant morphism

$$ \mathbb{F}_{\ell}[2d] \to A|_{\mathcal{M}_{\bar{W}}^{\ge N}} $$

inducing the given equivariant isomorphism $\mathbb{F}_{\ell}[2d] \xrightarrow{\sim} i^*A$. Let

$$ B \in D_{\text{ét}}(\mathcal{M}_{\bar{W}}^{\ge N}, \mathbb{F}_{\ell}) $$

be a cone of this morphism ans an $\pi^N$-equivariant object

$$ \mathbb{F}_{\ell}[2d] \to A|_{\mathcal{M}_{\bar{W}}^{\ge N}} \to B \xrightarrow{+1} . $$

Since $\mathbb{F}_{\ell}[2d]$ and $A$ (as the Verdier dual of $\mathbb{F}_{\ell}$, see the stability property of the ULA property under Verdier duality) are $g$-ULA, $B$ is $g$-ULA. In particular its Verdier dual is $g$-ULA and

$$ Rg!\mathbb{D}(B) $$

is constructible, where $\mathbb{D} = \mathbb{D}_{\mathcal{M}_{\bar{W}}^{\ge N}/S \times \pi^{\mathbb{N}} \ge N \cup \{+\infty\}}$. Since its fiber at $S = S \times \pi^{+\infty}$ is zero we deduce that, up to replacing $N$ by a bigger integer, we can suppose that

$$ Rg!\mathbb{D}(B) = 0. $$

Applying Verdier duality and using the biduality of $B$ we obtain

$$ Rg_*B = 0. $$

We not note $\Delta^{(n)}$ for $n \in \mathbb{N}_{\ge N}$ the fiber of $g$ over $S \times \pi^n$. The $\pi^N$-equivariance property implies that we have

$$ \Delta^{(n+1)} \subset \Delta^{(n)} $$

with

$$ \bigcap_{n \ge N} \Delta^{(n)} = s(S). $$

We thus have if $f^{(n)} : \Delta^{(n)} \to S$, and $C$ is the fiber of $B$ at $S \times \pi^N$: for all $n \ge N$

$$ Rf_*^{(n)} C|_{\mathcal{M}_Z^{(n)}} = 0. $$

We deduce that

$$ s^*C = \varinjlim_{n \ge N} Rf_*^{(n)} C = 0 $$

and thus

$$ \mathbb{F}_{\ell}[2d] \xrightarrow{\sim} s^*Rf^{!}\mathbb{F}_{\ell}. $$

This finishes the proof.
# 12th Lecture: August 15 2023

## 1. Local Shimura varieties

### 1.1. Definition.

1.1.1. The tower over the reflex field. Let $G$ over $E$ be as before. Let $[b] \in B(G)$ and $\{\mu\}$ be a conjugacy class cocharacters of $G_{\bar{E}}$ where we have fixed $\bar{E}$ an algebraic closure of $E$. Let $E_{\mu}|E$ be the field of definition of $\{\mu\}$, $E_{\mu} \subset \bar{E}$, and $\check{E}_{\mu}$ the completion of the maximal unramified extension of $E_{\mu}$. For $K \subset G(\mathbb{Q}_p)$ compact open let

$$ \mathcal{M}_K(G, b, \mu) \to \mathrm{Spa}(\check{E}_{\mu})^{\diamond} $$

be the associated local Shimura variety. If

$$ \mathcal{M}(G, b, \mu) = \lim_{\leftarrow K} \mathcal{M}_K(G, b, \mu), $$

The $v$-sheaf $\mathcal{M}(G, b, \mu)$ sends $S$ an $\overline{\mathbb{F}}_q$-perf. space to an untilt $S^{\sharp}$ over $\check{E}_{\mu}$ and an isomorphism

$$ \mathcal{E}_{1|X_S \setminus S^{\sharp}} \xrightarrow{\sim} \mathcal{E}_{b|X_S \setminus S^{\sharp}} $$

that is meromorphic along the Cartier divisor $S^{\sharp} \hookrightarrow X_S$ and is of type $\le \mu$ geometrically fiberwise over $S$.

This is representable by a locally spatial diamond compactifiable of finite dim. trg. over $\mathrm{Spa}(\check{E}_{\mu})$. There are two morphisms

<figure><img src="image_62.png" /></figure>

where

(1) $\mathcal{M}_K(G, b, \mu) = \underline{K} \setminus \mathcal{M}(G, b, \mu)$,

(2) $\mathrm{Gr}_G$ is the $B_{dR}$-affine Grassmanian and $\mathrm{Gr}^{\le \mu}$ the closed Schubert cell defined by $\mu$, a spatial diamond proper over $\mathrm{Spa}(\check{E}_{\mu})^{\diamond}$,
The geometric Satake correspondence allows us to define a perverse ULA (relative to the morphism toward $\mathrm{Div}_{E_\mu}^1$) sheaf

$$j_* \Lambda \in D_{\mathrm{ét}}(Hck^{\le \mu}, \Lambda)$$

where $j$ is the inclusion of the open Schubert cell. This corresponds to the representation of $\widehat{G} \times W_{E_\mu}$ with weight $\mu$.

By pull-back we obtain a on object

$$S_{\mu} \in D_{\mathrm{ét}}(\underline{[G(E) \times \widetilde{G}_b \setminus \mathrm{Sht}(G, b, \mu)]}, \Lambda)$$

## 1.2. Cohomology.

1.2.1. *The equivariant cohomology complex.* One has

$$\mathrm{Div}_{E_{\mu}}^{1} = \mathrm{Spa}(\check{E}_{\mu})^{\circ} / \varphi_{E_{\mu}}^{\mathbb{Z}} = [\mathrm{Spa}(\mathbb{C}_{p}^{\flat}) / \underline{W_{E_{\mu}}}]$$

There is thus a morphism

$$\mathrm{Div}_{E_{\mu}}^{1} \to [\underline{* / W_{E_{\mu}}}]$$

**LEMMA 1.1** (Drinfeld lemma; particular case). *Pull-back induces an equivalence*

$$D(G_b(E) \times W_{E_\mu}, \Lambda) \xrightarrow{\sim} D_{\mathrm{ét}}([\underline{* / \widetilde{G}_b}] \times [\underline{* / W_{E_\mu}}], \Lambda) \xrightarrow{\sim} D_{\mathrm{ét}}([\underline{* / \widetilde{G}_b}] \times \mathrm{Div}_{E_\mu}^1], \Lambda).$$

1.2.2. *The theorem.* Here is the theorem we want to prove. Let

$$f_K : [\widetilde{G}_b \setminus \mathrm{Sht}_K(G, b, \mu)] \to [\underline{* / \widetilde{G}_b}] \times \mathrm{Div}_{E_\mu}^1$$

**THEOREM 1.2.** *For all compact open subset $K$ of $G(E)$*

$$Rf_{K!} S_{\mu} \in D(G_b(E), \Lambda)^{BW_{E_{\mu}}}$$

is a compact object in $D(G_b(E), \Lambda)$ i.e. an object in the thick triangulated sub-category of $D(G_b(E), \Lambda)$ generated by the $c$-Ind$_{K'}^{G_b(E)} \Lambda$ when $K'$ goes through the set of compact open pro-$p$ subgroups of $G_b(E)$.

## 2. Local charts using the Jacobian criterion

**2.1. Construction of the local charts.** Let $[b] \in B(G)$. Suppose, to simplify that $G$ is quasi-split (if not everytuing works since $G \times X_S$ is a quasi-split group scheme over $X_S$). Let $[\nu_b] \in X_*(A)_Q^+$, $M_b$ its centralizer, a standard Levi, and $P_b$ the standard parabolic subgroup associated. We note $b_{M_b}$ for the canonical reduction of $b$ to $M_b$.
**THEOREM 2.2.** The v-sheaf $\tilde{\mathcal{M}}_b$ satisfies

(1) $\tilde{\mathcal{M}}_b \setminus \{\ast\}$ is a spatial diamond

(2) If $i : \{\ast\} \hookrightarrow \tilde{\mathcal{M}}_b$, for any $A \in D_{\acute{e}t}(\tilde{\mathcal{M}}_b, \Lambda)$ one has

$$R\Gamma(\tilde{\mathcal{M}}_b, A) \xrightarrow{\sim} i^*A.$$

**3. Some compact generators of $D_{\acute{e}t}(\mathrm{Bun}_G, \Lambda)$**

**DEFINITION 3.1.** For $[b] \in B(G)$ and $K$ a compact open pro-p subgroup of $G_b(E)$ define

$$A_K^b = Rf_{K!}^b Rf_K^{b!} \Lambda \in D_{\acute{e}t}(\mathrm{Bun}_G, \Lambda).$$

**EXAMPLE 3.2.** If $b$ is basic then

$$A_K^b = i_!^b(c - \mathrm{Ind}_K^{G_b(E)} \Lambda)$$

and thus $A_K^b$ corresponds in this case to the standard generator $c - \mathrm{Ind}_K^{G_b(E)} \Lambda$ of $D(G_b(E), \Lambda)$.

**PROPOSITION 3.3.** The following is satisfied

(1) For any $B \in D_{\acute{e}t}(\mathrm{Bun}_G, \Lambda)$ one has

$$\mathrm{Hom}(A_K^b, B) = (i^b)^* B.$$

(2) The collection $(A_K^b)_{[b],K}$ is a set of compact generators of $D_{\acute{e}t}(\mathrm{Bun}_G, \Lambda)$.

→ In particular,

$$D_{\acute{e}t}(\mathrm{Bun}_G, \Lambda)^\omega = \text{thick triangulated sub-cat. generated by } (A_K^b)_{[b],K}.$$

**4. The compactness criterion**

Here is the main theorem.

**THEOREM 4.1 (Compacity criterion).** An object $A \in D_{\acute{e}t}(\mathrm{Bun}_G, \Lambda)$ is compact iff $\{|b| \mid (i^b)^* A \neq 0\}$ is finite (i.e. $A$ is supported on a $qc.$ open subset of $\mathrm{Bun}_G$) and for all $[b] \in B(G)$,

$$(i^b)^* A \in D(G_b(E), \Lambda)$$

is compact (i.e. lies in the thick triangulated sub-category generated by the collection $(c - \mathrm{Ind}_K^{G_b(E)} \Lambda)_{K \subset G_b(E)}$). open pro-p
PROOF. Its is evident that A compact implies it is supported on a qc open subset. Let $U \subset \text{Bun}_G$ be such a qc open subset and $A \in D_{\text{ét}}(U, \Lambda)$. Choose $[b] \in |U|$ a closed point. Let $j: U \setminus \{[b]\} \to U$. There is an exact triangle

$$i_*^b i_*^{b*} A \to A \to j_* j^* A \xrightarrow{+1} .$$

The functors $i_*^b$ and $j_*$ have right adjoints that commute with arbitrary direct sums (in fact $R(i^b)^*$ is isomorphic to a shift of $(i^b)^*$ since $U$ and $[*/\tilde{G}_b]$ are $\ell$-cohomologically smooth). Thus, at the end we just need to prove, by induction on the cardinality of $|U|$, that if $A$ is compact then $j_*^* A$ is compact in $D_{\text{ét}}(U \setminus \{[b]\}, \Lambda)$. Since $D_{\text{ét}}(U, \Lambda)^\omega$ is the thick triangulated sub-category generated by the $A_{K'}^{b'}$ with $[b'] \in |U|$ suffices to prove that for $[b'] \in |U|$ and $K' \subset G_{b'}(E)$ open pro-$p$,

$$j^* A_{K'}^{b'}$$

is compact. If $[b'] \neq [b]$ one has $j_*^* A_{K'}^{[b']}$ and the result is evident. If $[b'] = [b]$ one has

$$j_*^* A_K^{[b]} = R f_{K'}^{b, \circ} R (f_K^{b, \circ})^! \Lambda$$

that is compact since $\tilde{M}_b^\circ := \tilde{M}_b \setminus \{*\}$ is spatial and thus quasi-compact. $\square$

→ the key point of this proof is to prove that $j^*$ sends compact objects to compact objects. This is absolutely not evident since $Rj_*^*$ does not commute with arbitrary direct sums in general since, as said before, $\text{Bun}_G$ being not quasi-separated, $j$ is not quasi-compact in general.

## 5. Stability of compact objects under the action of Hecke correspondences

**5.1. Hecke correspondences.** Let us consider the 2-category with one object whose 1-morphisms are $\text{Rep}_\Lambda(\hat{G})$ with composition given by $\otimes_\Lambda$ and whose 2-morphisms are the usual morphisms in $\text{Rep}_\Lambda(\hat{G})$. There is a morphism of 2-categories from this 2-category to $C_*$ the category of correspondences over $*$. This is given by the geometric Satake correspondence.

**5.2. Stability of compact objects.** The following result is formal.

**PROPOSITION 5.1.** For any $W \in \text{Rep}(\hat{G})$, $T_W \in \text{End}(D_{\text{ét}}(\text{Bun}_G, \Lambda))$ sends compact objects to compact objects;

$$T_W : D_{\text{ét}}(\text{Bun}_G, \Lambda)^\omega \to D_{\text{ét}}(\text{Bun}_G, \Lambda)^\omega.$$

PROOF. This is a consequence of the fact that

$$T : \text{Rep}(\hat{G}) \to \text{End}(D_{\text{ét}}(\text{Bun}_G, \Lambda))$$

is a monoidal functor, $T_{W_1 \otimes W_2} = T_{W_1} \circ T_{W_2}$. Thus, by application of $T$ to $1 \to \check{W} \otimes W$ and $\check{W} \otimes W \to 1$ on obtains that $T_{\check{W}}$ is a right adjoint to $T_W$. Since $T_{\check{W}}$ commutes with arbitrary direct sums we deduce the result. $\square$
6. Proof of the finitess result

It suffices to verify that

is identified with

$$Rf_{K!S_{\mu}} \in D(G_b(E), \Lambda)$$

$$(i^b)^* T_{\mu} \left( i_!^1 c - \mathrm{Ind}_K^{G(E)} \Lambda \right).$$
<figure><img src="image_63.png" /></figure>
Bibliography