More lemmas in etale-cohomology

etale-cohomology.tex

@@ -11855,21 +11855,6 @@ \section{Constructible sheaves}
 The proofs of (1) and (2) are very similar and are omitted.
 \end{proof}
 
-\begin{lemma}
-\label{lemma-tensor-product-constructible}
-Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring.
-The tensor product of two constructible sheaves of $\Lambda$-modules
-on $X_\etale$ is a constructible sheaf of $\Lambda$-modules.
-\end{lemma}
-
-\begin{proof}
-The question immediately reduces to the case where $X$ is affine.
-Since any two partitions of $X$ with constructible locally
-closed strata have a common refinement of the same type and
-since pullbacks commute with tensor product we reduce to
-Lemma \ref{lemma-tensor-product-locally-constant}.
-\end{proof}
-
 \begin{lemma}
 \label{lemma-support-constructible}
 Let $X$ be a quasi-compact and quasi-separated scheme.
@@ -11994,6 +11979,40 @@ \section{Constructible sheaves}
 factors through $\mathcal{F}$ as desired.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-tensor-product-constructible}
+Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring.
+The tensor product of two constructible sheaves of $\Lambda$-modules
+on $X_\etale$ is a constructible sheaf of $\Lambda$-modules.
+\end{lemma}
+
+\begin{proof}
+The question immediately reduces to the case where $X$ is affine.
+Since any two partitions of $X$ with constructible locally
+closed strata have a common refinement of the same type and
+since pullbacks commute with tensor product we reduce to
+Lemma \ref{lemma-tensor-product-locally-constant}.
+\end{proof}
+
+\begin{lemma}
+\label{lemma-tensor-constructible}
+Let $\Lambda \to \Lambda'$ be a homomorphism of Noetherian rings.
+Let $X$ be a scheme. Let $\mathcal{F}$ be a constructible
+sheaf of $\Lambda$-modules on $X_\etale$. Then
+$\mathcal{F} \otimes_{\underline{\Lambda}} \underline{\Lambda'}$
+is a constructible sheaf of $\Lambda'$-modules.
+\end{lemma}
+
+\begin{proof}
+Omitted. Hint: affine locally you can use the same stratification.
+\end{proof}
+
+
+
+
+
+
+
 
 
 
@@ -13345,6 +13364,78 @@ \section{Specializations and \'etale sheaves}
 We omit the proof that (1) implies (2).
 \end{proof}
 
+\begin{lemma}
+\label{lemma-characterize-locally-constant-module}
+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 $S_\etale$.
+The following are equivalent
+\begin{enumerate}
+\item $\mathcal{F}$ is a finite type, locally constant sheaf
+of $\Lambda$-modules, and
+\item 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.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The proof of this lemma is the same as the proof of
+Lemma \ref{lemma-characterize-locally-constant}.
+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 \'etale neighbourhood
+$(U, \overline{u})$ of $(S, \overline{s})$ such that $\mathcal{F}|_U$
+is constant. We may and do assume $S$ is affine.
+
+\medskip\noindent
+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 $U_\etale$ 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 \ref{remark-map-stalks}).
+The image $u \in U$ of $\overline{u}$ is in $E$.
+By our assumption on the specialization maps for $\mathcal{F}$,
+by Remark \ref{remark-can-lift}, and by
+Lemma \ref{lemma-specialization-map-pullback}
+we see that $E$ is closed under specializations
+and generalizations in the topological space $U$.
+
+\medskip\noindent
+After shrinking $U$ we may assume $U$ is affine too. By
+Descent, Lemma \ref{descent-lemma-locally-finite-nr-irred-local-fppf}
+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 \ref{theorem-exactness-stalks}. Thus (1) follows.
+
+\medskip\noindent
+We omit the proof that (1) implies (2).
+\end{proof}
+
 \begin{lemma}
 \label{lemma-specialization-map-pushforward}
 Let $f : X \to S$ be a quasi-compact and quasi-separated morphism of schemes.
@@ -18643,36 +18734,47 @@ \section{Local acyclicity}
 
 \begin{lemma}
 \label{lemma-locally-acyclic-locally-constant}
-Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a finite
+Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a
 locally constant abelian sheaf on $X_\etale$ such that for every geometric
-point $\overline{x}$ of $X$ the finite abelian group
-$\mathcal{F}_{\overline{x}}$ has order prime to the characteristic of the
+point $\overline{x}$ of $X$ the abelian group
+$\mathcal{F}_{\overline{x}}$ is a torsion group all of whose elements have
+order prime to the characteristic of the
 residue field of $\overline{x}$. If $f$ is locally acyclic, then $f$ is
 locally acyclic relative to $\mathcal{F}$.
 \end{lemma}
 
 \begin{proof}
 Namely, let $\overline{x}$ be a geometric point of $X$.
-Since $\mathcal{F}$ is finite locally constant we see that
+Since $\mathcal{F}$ is locally constant we see that
 the restriction of $\mathcal{F}$ to $\Spec(\mathcal{O}^{sh}_{X, \overline{x}})$
 is isomorphic to the constant sheaf $\underline{M}$ with
 $M = \mathcal{F}_{\overline{x}}$. By assumption we can write
-$M = \bigoplus \mathbf{Z}/n_i\mathbf{Z}$ for some integers $n_i$
-prime to the characteristic of the residue field of $\overline{x}$.
+$M = \colim M_i$ as a filtered colimit of finite abelian groups
+$M_i$ of order prime to the characteristic of the residue field of
+$\overline{x}$. Consider a geometric point $\overline{t}$ of
+$\Spec(\mathcal{O}^{sh}_{S, f(\overline{x})})$.
+Since $F_{\overline{x}, \overline{t}}$ is affine, we have
+$$
+H^q(F_{\overline{x}, \overline{t}}, \underline{M}) =
+\colim H^q(F_{\overline{x}, \overline{t}}, \underline{M_i})
+$$
+by Lemma \ref{lemma-colimit}.
+For each $i$ we can write $M_i = \bigoplus \mathbf{Z}/n_{i, j}\mathbf{Z}$
+as a finite direct sum for some integers $n_{i, j}$ prime to the
+characteristic of the residue field of $\overline{x}$.
 Since $f$ is locally acyclic we see that
 $$
-H^q(F_{\overline{x}, \overline{t}}, \underline{\mathbf{Z}/n_i\mathbf{Z}}) =
+H^q(F_{\overline{x}, \overline{t}},
+\underline{\mathbf{Z}/n_{i, j}\mathbf{Z}}) =
 \left\{
 \begin{matrix}
-\mathbf{Z}/n_i\mathbf{Z} & \text{if} & q = 0 \\
+\mathbf{Z}/n_{i, j}\mathbf{Z} & \text{if} & q = 0 \\
 0 & \text{if} & q \not = 0
 \end{matrix}
 \right.
 $$
-for any geometric point $\overline{t}$ of
-$\Spec(\mathcal{O}^{sh}_{S, f(\overline{x})})$.
 See discussion following Definition \ref{definition-locally-acyclic}.
-Taking direct sums we conclude that
+Taking the direct sums and the colimit we conclude that
 $$
 H^q(F_{\overline{x}, \overline{t}}, \underline{M}) =
 \left\{
@@ -18959,6 +19061,29 @@ \section{The cospecialization map}
 and we conclude.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-sp-isom-proper-loc-cst-torsion}
+Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$
+be an abelian sheaf on $X_\etale$. Assume
+\begin{enumerate}
+\item $f$ is smooth and proper
+\item $\mathcal{F}$ is locally constant, and
+\item $\mathcal{F}_{\overline{x}}$ is a torsion group all of
+whose elements have order prime to the residue characteristic of
+$\overline{x}$ for every geometric point $\overline{x}$ of $X$.
+\end{enumerate}
+Then for every geometric point $\overline{s}$ of $S$ and every geometric
+point $\overline{t}$ of $\Spec(\mathcal{O}^{sh}_{S, \overline{s}})$
+the specialization map
+$sp : (Rf_*\mathcal{F})_{\overline{s}} \to (Rf_*\mathcal{F})_{\overline{t}}$
+is an isomorphism.
+\end{lemma}
+
+\begin{proof}
+This follows from Lemmas \ref{lemma-sp-isom-proper-torsion-loc-ac}
+and \ref{lemma-locally-acyclic-locally-constant} and
+Proposition \ref{proposition-smooth-locally-acyclic}.
+\end{proof}