59.75 Specializations and étale sheaves
Topological picture: Let $X$ be a topological space and let $x' \leadsto x$ be a specialization of points in $X$. Then every open neighbourhood of $x$ contains $x'$. Hence for any sheaf $\mathcal{F}$ on $X$ there is a specialization map
\[ sp : \mathcal{F}_ x \longrightarrow \mathcal{F}_{x'} \]
of stalks sending the equivalence class of the pair $(U, s)$ in $\mathcal{F}_ x$ to the equivalence class of the pair $(U, s)$ in $\mathcal{F}_{x'}$; see Sheaves, Section 6.11 for the description of stalks in terms of equivalence classes of pairs. Of course this map is functorial in $\mathcal{F}$, i.e., $sp$ is a transformation of functors.
For sheaves in the étale topology we can mimick this construction, see [Exposee VII, 7.7, page 397, SGA4]. To do this suppose we have a scheme $S$, a geometric point $\overline{s}$ of $S$, and a geometric point $\overline{t}$ of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. For any sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ we will construct the specialization map
\[ sp : \mathcal{F}_{\overline{s}} \longrightarrow \mathcal{F}_{\overline{t}} \]
Here we have abused language: instead of writing $\mathcal{F}_{\overline{t}}$ we should write $\mathcal{F}_{p(\overline{t})}$ where $p : \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \to S$ is the canonical morphism. Recall that
\[ \mathcal{F}_{\overline{s}} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{F}(U) \]
where the colimit is over all étale neighbourhoods $(U, \overline{u})$ of $(S, \overline{s})$, see Section 59.29. Since $\mathcal{O}^{sh}_{S, \overline{s}}$ is the stalk of the structure sheaf, we find for every étale neighbourhood $(U, \overline{u})$ of $(S, \overline{s})$ a canonical map $\mathcal{O}_{U, u} \to \mathcal{O}^{sh}_{S, \overline{s}}$. Hence we get a unique factorization
\[ \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}) \to U \to S \]
If $\overline{v}$ denotes the image of $\overline{t}$ in $U$, then we see that $(U, \overline{v})$ is an étale neighbourhood of $(S, \overline{t})$. This construction defines a functor from the category of étale neighbourhoods of $(S, \overline{s})$ to the category of étale neighbourhoods of $(S, \overline{t})$. Thus we may define the map $sp : \mathcal{F}_{\overline{s}} \to \mathcal{F}_{\overline{t}}$ by sending the equivalence class of $(U, \overline{u}, \sigma )$ where $\sigma \in \mathcal{F}(U)$ to the equivalence class of $(U, \overline{v}, \sigma )$.
Let $K \in D(S_{\acute{e}tale})$. With $\overline{s}$ and $\overline{t}$ as above we have the specialization map
\[ sp : K_{\overline{s}} \longrightarrow K_{\overline{t}} \quad \text{in}\quad D(\textit{Ab}) \]
Namely, if $K$ is represented by the complex $\mathcal{F}^\bullet $ of abelian sheaves, then we simply that the map
\[ K_{\overline{s}} = \mathcal{F}^\bullet _{\overline{s}} \longrightarrow \mathcal{F}^\bullet _{\overline{t}} = K_{\overline{t}} \]
which is termwise given by the specialization maps for sheaves constructed above. This is independent of the choice of complex representing $K$ by the exactness of the stalk functors (i.e., taking stalks of complexes is well defined on the derived category).
Clearly the construction is functorial in the sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$. If we think of the stalk functors as morphisms of topoi $\overline{s}, \overline{t} : \textit{Sets} \to \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$, then we may think of $sp$ as a $2$-morphism
\[ \xymatrix{ \textit{Sets} \rrtwocell ^{\overline{t}}_{\overline{s}}{\ sp} & & \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) } \]
of topoi.
Lemma 59.75.4. Let $g : S' \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a sheaf on $S_{\acute{e}tale}$. Let $\overline{s}'$ be a geometric point of $S'$, and let $\overline{t}'$ be a geometric point of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S', \overline{s}'})$. Denote $\overline{s} = g(\overline{s}')$ and $\overline{t} = h(\overline{t}')$ where $h : \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S', \overline{s}'}) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$ is the canonical morphism. For any sheaf $\mathcal{F}$ on $S_{\acute{e}tale}$ the specialization map
\[ sp : (f^{-1}\mathcal{F})_{\overline{s}'} \longrightarrow (f^{-1}\mathcal{F})_{\overline{t}'} \]
is equal to the specialization map $sp : \mathcal{F}_{\overline{s}} \to \mathcal{F}_{\overline{t}}$ via the identifications $(f^{-1}\mathcal{F})_{\overline{s}'} = \mathcal{F}_{\overline{s}}$ and $(f^{-1}\mathcal{F})_{\overline{t}'} = \mathcal{F}_{\overline{t}}$ of Lemma 59.36.2.
Proof.
Omitted.
$\square$
Lemma 59.75.5. Let $S$ be a scheme such that every quasi-compact open of $S$ has finite number of irreducible components (for example if $S$ has a Noetherian underlying topological space, or if $S$ is locally Noetherian). Let $\mathcal{F}$ be a sheaf of sets on $S_{\acute{e}tale}$. The following are equivalent
$\mathcal{F}$ is finite locally constant, and
all stalks of $\mathcal{F}$ are finite sets and all specialization maps $sp : \mathcal{F}_{\overline{s}} \to \mathcal{F}_{\overline{t}}$ are bijective.
Proof.
Assume (2). Let $\overline{s}$ be a geometric point of $S$ lying over $s \in S$. In order to prove (1) we have to find an étale neighbourhood $(U, \overline{u})$ of $(S, \overline{s})$ such that $\mathcal{F}|_ U$ is constant. We may and do assume $S$ is affine.
Since $\mathcal{F}_{\overline{s}}$ is finite, we can choose $(U, \overline{u})$, $n \geq 0$, and pairwise distinct elements $\sigma _1, \ldots , \sigma _ n \in \mathcal{F}(U)$ such that $\{ \sigma _1, \ldots , \sigma _ n\} \subset \mathcal{F}(U)$ maps bijectively to $\mathcal{F}_{\overline{s}}$ via the map $\mathcal{F}(U) \to \mathcal{F}_{\overline{s}}$. Consider the map
\[ \varphi : \underline{\{ 1, \ldots , n\} } \longrightarrow \mathcal{F}|_ U \]
on $U_{\acute{e}tale}$ defined by $\sigma _1, \ldots , \sigma _ n$. This map is a bijection on stalks at $\overline{u}$ by construction. Let us consider the subset
\[ E = \{ u' \in U \mid \varphi _{\overline{u}'}\text{ is bijective}\} \subset U \]
Here $\overline{u}'$ is any geometric point of $U$ lying over $u'$ (the condition is independent of the choice by Remark 59.29.8). The image $u \in U$ of $\overline{u}$ is in $E$. By our assumption on the specialization maps for $\mathcal{F}$, by Remark 59.75.3, and by Lemma 59.75.4 we see that $E$ is closed under specializations and generalizations in the topological space $U$.
After shrinking $U$ we may assume $U$ is affine too. By Descent, Lemma 35.16.3 we see that $U$ has a finite number of irreducible components. After removing the irreducible components which do not pass through $u$, we may assume every irreducible component of $U$ passes through $u$. Since $U$ is a sober topological space it follows that $E = U$ and we conclude that $\varphi $ is an isomorphism by Theorem 59.29.10. Thus (1) follows.
We omit the proof that (1) implies (2).
$\square$
Lemma 59.75.6. Let $S$ be a scheme such that every quasi-compact open of $S$ has finite number of irreducible components (for example if $S$ has a Noetherian underlying topological space, or if $S$ is locally Noetherian). Let $\Lambda $ be a Noetherian ring. Let $\mathcal{F}$ be a sheaf of $\Lambda $-modules on $S_{\acute{e}tale}$. The following are equivalent
$\mathcal{F}$ is a finite type, locally constant sheaf of $\Lambda $-modules, and
all stalks of $\mathcal{F}$ are finite $\Lambda $-modules and all specialization maps $sp : \mathcal{F}_{\overline{s}} \to \mathcal{F}_{\overline{t}}$ are bijective.
Proof.
The proof of this lemma is the same as the proof of Lemma 59.75.5. Assume (2). Let $\overline{s}$ be a geometric point of $S$ lying over $s \in S$. In order to prove (1) we have to find an étale neighbourhood $(U, \overline{u})$ of $(S, \overline{s})$ such that $\mathcal{F}|_ U$ is constant. We may and do assume $S$ is affine.
Since $M = \mathcal{F}_{\overline{s}}$ is a finite $\Lambda $-module and $\Lambda $ is Noetherian, we can choose a presentation
\[ \Lambda ^{\oplus m} \xrightarrow {A} \Lambda ^{\oplus n} \to M \to 0 \]
for some matrix $A = (a_{ji})$ with coefficients in $\Lambda $. We can choose $(U, \overline{u})$ and elements $\sigma _1, \ldots , \sigma _ n \in \mathcal{F}(U)$ such that $\sum a_{ji}\sigma _ i = 0$ in $\mathcal{F}(U)$ and such that the images of $\sigma _ i$ in $\mathcal{F}_{\overline{s}} = M$ are the images of the standard basis element of $\Lambda ^ n$ in the presentation of $M$ given above. Consider the map
\[ \varphi : \underline{M} \longrightarrow \mathcal{F}|_ U \]
on $U_{\acute{e}tale}$ defined by $\sigma _1, \ldots , \sigma _ n$. This map is a bijection on stalks at $\overline{u}$ by construction. Let us consider the subset
\[ E = \{ u' \in U \mid \varphi _{\overline{u}'}\text{ is bijective}\} \subset U \]
Here $\overline{u}'$ is any geometric point of $U$ lying over $u'$ (the condition is independent of the choice by Remark 59.29.8). The image $u \in U$ of $\overline{u}$ is in $E$. By our assumption on the specialization maps for $\mathcal{F}$, by Remark 59.75.3, and by Lemma 59.75.4 we see that $E$ is closed under specializations and generalizations in the topological space $U$.
After shrinking $U$ we may assume $U$ is affine too. By Descent, Lemma 35.16.3 we see that $U$ has a finite number of irreducible components. After removing the irreducible components which do not pass through $u$, we may assume every irreducible component of $U$ passes through $u$. Since $U$ is a sober topological space it follows that $E = U$ and we conclude that $\varphi $ is an isomorphism by Theorem 59.29.10. Thus (1) follows.
We omit the proof that (1) implies (2).
$\square$
Lemma 59.75.7. Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $K \in D^+(X_{\acute{e}tale})$. Let $\overline{s}$ be a geometric point of $S$ and let $\overline{t}$ be a geometric point of $\mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}})$. We have a commutative diagram
\[ \xymatrix{ (Rf_*K)_{\overline{s}} \ar[r]_{sp} \ar@{=}[d] & (Rf_*K)_{\overline{t}} \ar@{=}[d] \\ R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{s}}), K) \ar[r] & R\Gamma (X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}), K) } \]
where the bottom horizontal arrow arises as pullback by the morphism $\text{id}_ X \times c$ where $c : \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{t}}) \to \mathop{\mathrm{Spec}}(\mathcal{O}^{sh}_{S, \overline{S}})$ is the morphism introduced in Remark 59.75.2. The vertical arrows are given by Theorem 59.53.1.
Proof.
This follows immediately from the description of $sp$ in Remark 59.75.2.
$\square$
Comments (1)
Comment #8639 by Emmanuel Kowalski on