Improve a proof

Thanks to Michael Barz https://stacks.math.columbia.edu/tag/094Q#comment-8542
stacks · May 13, 2024 · d715958 · d715958
1 parent 4b53ec5
commit d715958
Show file tree

Hide file tree

Showing 2 changed files with 4 additions and 6 deletions.
diff --git a/CONTRIBUTORS b/CONTRIBUTORS
@@ -28,6 +28,7 @@ Guillermo Barajas Ayuso
 Adrian Barquero-Sanchez
 Owen Barrett
 Reid Barton
+Michael Barz
 Tathagata Basak
 Giulia Battiston
 Jefferson Baudin

diff --git a/more-morphisms.tex b/more-morphisms.tex
@@ -18758,12 +18758,9 @@ \section{Weakly \'etale morphisms}
 \mathcal{O}_{X, x}
 $$
 where the first map is a localization (hence flat) and the second map is a
-surjection (hence an epimorphism of rings).
-Condition (1) means that for all $x$ the second arrow is flat.
-Condition (2) is that for all $x$ the composition is flat.
-These conditions are equivalent by
-Algebra, Lemma \ref{algebra-lemma-composition-flat} and
-More on Algebra, Lemma \ref{more-algebra-lemma-key}.
+surjection. Condition (1) means that the second arrow is flat for all $x$.
+Condition (2) is that the composition is flat for all $x$. Thus the equivalence
+by Algebra, Lemma \ref{algebra-lemma-flat-localization} part (2).
 \end{proof}
 
 \begin{lemma}