stacks/stacks-project

Fix statement of two lemmas

These lemmas can perhaps be moved to the obsolete chapter...
 \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}) =