# stacks/stacks-project

Application of smooth base change

 @@ -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