lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -974,16 +974,16 @@ be easyer to check than Ore's -- known to imply Ore's condition. For
 instance\dots
 
 \begin{lemma}
-  Let \(S \subset R\) be a multiplicative subset generated by locally
-  \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\) such
-  that for each \(r \in R\) there exists \(n > 0\) such that
+  Let \(S \subset R\) be a multiplicative subset generated by finitely many
+  locally \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\)
+  such that for each \(r \in R\) there exists \(n > 0\) such that
   \(\operatorname{ad}(s)^n r = [s, [s, \cdots [s, r]]\cdots] = 0\). Then \(S\)
   satisfies Ore's localization condition.
 \end{lemma}
 
-In our case, we are more interested in formally inverting the action of the
-action of \(F_\alpha\) in \(V\) than in inverting \(F_\alpha\) itself. To that
-end, we introduce one further construction, kwon as \emph{the localization of a
+In our case, we are more interested in formally inverting the action of
+\(F_\alpha\) in \(V\) than in inverting \(F_\alpha\) itself. To that end, we
+introduce one further construction, kwon as \emph{the localization of a
 module}.
 
 \begin{definition}
@@ -1037,8 +1037,9 @@ well-behaved. For example, we can show\dots
 \end{lemma}
 
 \begin{note}
-  The basis \(\Sigma\) may very well depend on the representation \(V\)! This
-  is another obstacle to showing the functoriality of our constructions.
+  The basis \(\Sigma\) in lemma~\ref{thm:nice-basis-for-inversion} may very
+  well depend on the representation \(V\)! This is another obstruction to the
+  functoriality of our constructions.
 \end{note}
 
 Since \(F_\alpha\) is locally \(\operatorname{ad}\)-nilpotent for all \(\alpha