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Improve formulation of a lemma
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aisejohan committed Aug 14, 2022
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Expand Up @@ -3649,12 +3649,13 @@ \section{Weil cohomology theories, II}
$H^0(X) \to H^0(x_1) \oplus \ldots \oplus H^0(x_r)$ is injective,
\item the map $H^0(\Spec(k')) \to H^0(X)$ is an isomorphism.
\end{enumerate}
If $X$ is equidimensional of dimension $d$, these are also equivalent to
If this is true, then $H^0(X)$ is a finite separable algebra over $F$.
If $X$ is equidimensional of dimension $d$, then (1) and (2)
are also equivalent to
\begin{enumerate}
\item[(3)] the classes of closed points generate $H^{2d}(X)(d)$
as a module over $H^0(X)$.
\end{enumerate}
If this is true, then $H^0(X)$ is a finite separable algebra over $F$.
\end{lemma}

\begin{proof}
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