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Fix statement of two lemmas

These lemmas can perhaps be moved to the obsolete chapter...
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aisejohan committed Nov 6, 2018
1 parent 6799d95 commit 1a4dc4435005962ccdf2fa3a876b31b39415b8f1
Showing with 13 additions and 4 deletions.
  1. +13 −4 etale-cohomology.tex

\begin{lemma}
\label{lemma-base-change-f-star-general-stalks}
Let $f : X \to S$ be a morphism of schemes. Let $g : T \to S$ be a
quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be an
Consider a cartesian diagram of schemes
$$
\xymatrix{
X \ar[d]_f & Y \ar[l]^h \ar[d]^e \\
S & T \ar[l]_g
}
$$
where $g : T \to S$ is quasi-compact and quasi-separated.
Let $\mathcal{F}$ be an
abelian sheaf on $T_\etale$. Let $q \geq 0$. The following are equivalent
\begin{enumerate}
\item For every geometric point $\overline{x}$ of $X$ with image
$\Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_X Y =
\Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_S T$. Thus
the map in (1) is the map of stalks at $\overline{x}$ for the map
in (2) by Theorem \ref{theorem-higher-direct-images}.
in (2) by Theorem \ref{theorem-higher-direct-images} (and
Lemma \ref{lemma-stalk-pullback}).
Thus the result by Theorem \ref{theorem-exactness-stalks}.
\end{proof}

Let $\overline{x}$ be a geometric point of $X$ with image $\overline{s}$ in $S$.
Let $\Spec(K) \to \Spec(\mathcal{O}^{sh}_{S, \overline{s}})$
be a morphism with $K$ a separably closed field. Let $\mathcal{F}$ be an
abelian sheaf on $T_\etale$. Let $q \geq 0$. The following are equivalent
abelian sheaf on $\Spec(K)_\etale$. Let $q \geq 0$. The following are
equivalent
\begin{enumerate}
\item
$H^q(\Spec(\mathcal{O}^{sh}_{X, \overline{x}}) \times_S \Spec(K), \mathcal{F}) =

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