lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -158,7 +158,7 @@
 
 % TODO: Add an example: there's an example of a coherent sl2-family in
 % Mathieu's paper
-% TODOO: Add a discussion on how this may sound unintuitive, but the motivation
+% TODO: Add a discussion on how this may sound unintuitive, but the motivation
 % comes from the relationship between highest weight modules and coherent
 % families
 
@@ -170,9 +170,9 @@
 \end{definition}
 
 \begin{definition}
-  Given a representation \(V\) of \(\mathfrak{g}\), a coherent extension
-  \(\mathcal{M}\) of \(V\) is a coherent family \(\mathcal{M}\) that contains
-  \(V\) as a subquotient.
+  Given an admissible representation \(V\) of \(\mathfrak{g}\) of degree \(d\),
+  a coherent extension \(\mathcal{M}\) of \(V\) is a coherent family
+  \(\mathcal{M}\) of degree \(d\) that contains \(V\) as a subquotient.
 \end{definition}
 
 % Mathieu's proof of this is somewhat profane, I don't think it's worth
@@ -283,7 +283,6 @@
   s^{-1} = t_2^{-1} u_2\).
 \end{definition}
 
-% TODOOO: Does R need to be Noetherian?
 \begin{theorem}[Ore-Asano]
   Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
   condition. Then there exists a (unique) ring \(S^{-1} R\), with a canonical
@@ -440,11 +439,13 @@
       \Sigma^{-1} V\) is the \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
       \(\Sigma^{-1} V\) twisted by the automorphism \(\theta_\lambda\) then
       \(\Sigma^{-1} V_\mu = (\theta_\lambda \Sigma^{-1} V)_{\mu + \lambda}\).
+      In particular, \(\operatorname{supp} \theta_\lambda \Sigma^{-1} V =
+      \lambda + Q + \operatorname{supp} V\).
   \end{enumerate}
 \end{proposition}
 
 \begin{proposition}[Mathieu]
-  There exists a coherent extension \(\mathcal{M}\) of \(V\) with degree \(d\).
+  There exists a coherent extension \(\mathcal{M}\) of \(V\).
 \end{proposition}
 
 \begin{proof}
@@ -480,12 +481,10 @@
 \end{theorem}
 
 % TODOOO: Prove the uniqueness
-% TODOO: Does every coherent extension of V have the same degree as V?
 \begin{proof}
   The existence part should be clear from the previous discussion: let
-  \(\mathcal{M}\) be a coherent extension of \(V\) with \(\deg \mathcal{M} =
-  d\) and take \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of
-  \(V\). 
+  \(\mathcal{M}\) be a coherent extension of \(V\) and take
+  \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of \(V\). 
 
   We claim \(V\) is contained in \(\operatorname{Ext}(V)\). Indeed, if we fix
   some composition series \(0 =