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Application of smooth base change

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aisejohan committed Oct 8, 2018
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@@ -15478,11 +15478,90 @@ \section{Smooth base change}
Since $\xi$ dies in $X'_K$ this finishes the proof.
\end{proof}
\noindent
The following immediate consquence of the smooth base change
theorem is what is often used in practice.
\begin{lemma}
\label{lemma-smooth-base-change-general}
Let $S$ be a scheme. Let $S' = \lim S_i$ be a directed inverse
limit of schemes $S_i$ smooth over $S$ with affine transition
morphisms. Let $f : X \to S$ be quas-compact and quasi-separated
and for the fibre square
$$
\xymatrix{
X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\
S' \ar[r]^g & S
}
$$
Then
$$
g^{-1}Rf_*E = R(f')_*(g')^{-1}E
$$
for any $E \in D^+(X_\etale)$ whose cohomology sheaves $H^q(E)$
have stalks which are torsion of orders invertible on $S$.
\end{lemma}
\begin{proof}
Consider the spectral sequences
$$
E_2^{p, q} = R^pf_*H^q(E)
\quad\text{and}\quad
{E'}_2^{p, q} = R^pf'_*H^q((g')^{-1}E) = R^pf'_*(g')^{-1}H^q(E)
$$
converging to $R^nf_*E$ and $R^nf'_*(g')^{-1}E$.
These spectral sequences are constructed in
Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}.
Combining the smooth base change theorem
(Theorem \ref{theorem-smooth-base-change})
with Lemma \ref{lemma-base-change-Rf-star-colim} we see that
$$
g^{-1}R^pf_*H^q(E) = R^p(f')_*(g')^{-1}H^q(E)
$$
Combining all of the above we get the lemma.
\end{proof}
\section{Applications of smooth base change}
\label{section-applications-smooth-base-change}
\noindent
In this section we discuss some more or less immediate
consequences of the smooth base change theorem.
\begin{lemma}
\label{lemma-smooth-base-change-separably-closed}
Let $K/k$ be an extension of separably closed fields. Let $X$
be a scheme over $k$. Let $E \in D^+(X_\etale)$ have cohomology
sheaves whose stalks are torsion of orders invertible in $k$.
Then the map
$H^q_\etale(X, E) \to H^q_\etale(X_K, E|_{X_K})$
is an isomorphism for $q \geq 0$.
\end{lemma}
\begin{proof}
First let $\overline{k}$ and $\overline{K}$ be the algebraic closures
of $k$ and $K$. The morphisms $\Spec(\overline{k}) \to \Spec(k)$ and
$\Spec(\overline{K}) \to \Spec(K)$ are universal homeomorphisms
as $\overline{k}/k$ and $\overline{K}/K$ are purely inseparable
(see Algebra, Lemma \ref{algebra-lemma-p-ring-map}).
Thus $H^q_\etale(X, \mathcal{F}) =
H^q_\etale(X_{\overline{k}}, \mathcal{F}_{X_{\overline{k}}})$ by
the topological invariance of \'etale cohomology, see
Proposition \ref{proposition-topological-invariance}.
Similarly for $X_K$ and $X_{\overline{K}}$.
Thus we may assume $k$ and $K$ are algebraically closed.
In this case $K$ is a limit of smooth $k$-algebras, see
Algebra, Lemma \ref{algebra-lemma-colimit-syntomic}.
We conclude our lemma is a special case of
Theorem \ref{theorem-smooth-base-change} as reformulated in
Lemma \ref{lemma-smooth-base-change-general}.
\end{proof}
@@ -16088,31 +16167,34 @@ \section{The proper base change theorem}
The second one is a special case of Lemma \ref{lemma-proper-base-change}.
\end{proof}
\section{Applications of proper base change}
\label{section-applications-proper-base-change}
\noindent
In this section we discuss some more or less immediate
consequences of the proper base change theorem.
\begin{lemma}
\label{lemma-base-change-separably-closed}
Let $k \subset k'$ be an extension of separably closed fields.
Let $X$ be a proper scheme over $k$.
Let $\mathcal{F}$ be a torsion abelian sheaf on $X$.
Let $\mathcal{F}$ be a torsion abelian sheaf on $X_\etale$.
Then the map $H^q_\etale(X, \mathcal{F}) \to
H^q_\etale(X_{k'}, \mathcal{F}|_{X_{k'}})$ is an isomorphism
for $q \geq 0$.
\end{lemma}
\begin{proof}
This is a special case of Theorem \ref{theorem-proper-base-change}.
Looking at stalks we see that
this is a special case of
Theorem \ref{theorem-proper-base-change}.
\end{proof}
\section{Applications of proper base change}
\label{section-applications-proper-base-change}
\noindent
As an application of the proper base change theorem we obtain the following.
\begin{lemma}
\label{lemma-cohomological-dimension-proper}
Let $f : X \to Y$ be a proper morphism of schemes

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