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Replace F by F|_U

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aisejohan committed Sep 2, 2019
1 parent 53158d9 commit 67a8339e0da109fd891f191dd922580516d914fe
Showing with 4 additions and 4 deletions.
  1. +4 −4 sites-modules.tex
@@ -4428,18 +4428,18 @@ \section{Flat modules}
assume there exists a presentation
$$
\mathcal{O}_U^{\oplus r} \to
\mathcal{O}_U^{\oplus n} \to \mathcal{F} \to 0
\mathcal{O}_U^{\oplus n} \to \mathcal{F}|_U \to 0
$$
By Lemma \ref{lemma-flat-eq} we may assume, after replacing $U$
By Lemma \ref{lemma-flat-eq} we may, after replacing $U$
by the members of a covering, assume there exists a factorization
$$
\mathcal{O}_U^{\oplus n} \to
\mathcal{O}_U^{\oplus n_1} \to \mathcal{F}
\mathcal{O}_U^{\oplus n_1} \to \mathcal{F}|_U
$$
such that the composition
$\mathcal{O}_U^{\oplus r} \to \mathcal{O}_U^{\oplus n}
\to \mathcal{O}_U^{\oplus n_r}$ is zero.
This means that the surjection $\mathcal{O}_U^{\oplus n_r} \to \mathcal{F}$
This means that the surjection $\mathcal{O}_U^{\oplus n_r} \to \mathcal{F}|_U$
has a section and we win.
\end{proof}

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