lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -279,7 +279,7 @@
 \begin{corollary}
   Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
   \((F_\alpha : \alpha \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
-  the multiplicative subset generated by \(F_\alpha\), \(\alpha \in F_\alpha\).
+  the multiplicative subset generated by \(F_\alpha\), \(\alpha \in \Sigma\).
   The \(K\)-algebra \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
   \Sigma)}\) is well defined and the localization map
   \begin{align*}
@@ -341,6 +341,7 @@
   \(\operatorname{Ext}(V)\) is irreducible as a coherent family.
 \end{theorem}
 
+% TODOOO: Prove the uniqueness
 \begin{proof}
   The existence part should now be clear from the previous discussion: let
   \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
@@ -381,7 +382,7 @@
     \subset \operatorname{Ext}(V)
   \]
 
-  % TODOO: Prove that the weight spaces of any simple g-module are all simple
+  % TODOOO: Prove that the weight spaces of any simple g-module are all simple
   % C(h)-modules
   Since the degree of \(V\) is the same as the degree of
   \(\operatorname{Ext}(V)\), some of its weight spaces have maximal dimension
@@ -390,8 +391,6 @@
   a simple \(C_{\mathcal{U}(\mathfrak{h})}(\mathfrak{h})\)-module for some
   \(\lambda \in \operatorname{supp} V\), so that \(\operatorname{Ext}(V)\) is
   irreducible as a coherent family.
-
-  % TODOO: Prove the uniqueness
 \end{proof}
 
 \begin{proposition}[Mathieu]