lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -41,24 +41,23 @@
   \]
   is a weight module. It is clear that \(\mfrac{V_\lambda}{W} \subset
   \left(\mfrac{V}{W}\right)_\lambda\). To see that \(\mfrac{V_\lambda}{W} =
-  \left(\mfrac{V}{W}\right)_\lambda\), we remark that \(V_\lambda \cong V
-  \otimes_{\mathcal{U}(\mathfrak{h})}
-  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}\) as
-  \(\mathfrak{h}\)-modules, where \(\mathfrak{m}_\lambda \normal
-  \mathcal{U}(\mathfrak{h})\) is the left ideal generated by the elements \(H -
-  \lambda(H)\), \(H \in \mathfrak{h}\). Likewise
-  \(\left(\mfrac{V}{W}\right)_\lambda \cong \mfrac{V}{W}
-  \otimes_{\mathcal{U}(\mathfrak{h})}
-  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}\) and the diagram
+  \left(\mfrac{V}{W}\right)_\lambda\), we remark that \(V_\lambda \cong
+  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+  \otimes_{\mathcal{U}(\mathfrak{h})} V\) as \(\mathfrak{h}\)-modules, where
+  \(\mathfrak{m}_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal
+  generated by the elements \(H - \lambda(H)\), \(H \in \mathfrak{h}\).
+  Likewise \(\left(\mfrac{V}{W}\right)_\lambda \cong
+  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+  \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}\) and the diagram
   \begin{center}
     \begin{tikzcd}
       V_\lambda \arrow{d} \arrow{r}{\pi} &
       \left(\mfrac{V}{W}\right)_\lambda \arrow{d} \\
-      V \otimes_{\mathcal{U}(\mathfrak{h})}
       \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+      \otimes_{\mathcal{U}(\mathfrak{h})} V
       \arrow[swap]{r}{\pi \otimes \operatorname{id}} &
-      \mfrac{V}{W} \otimes_{\mathcal{U}(\mathfrak{h})}
       \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+      \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{V}{W}
     \end{tikzcd}
   \end{center}
   commutes, so that the projection \(V_\lambda \to
@@ -266,7 +265,7 @@
   \end{enumerate}
 \end{theorem}
 
-\section{Localizations \& Existance of Coherent Extensions}
+\section{Localizations \& the Existance of Coherent Extensions}
 
 % TODO: Comment on the intuition behind the proof: we can get vectors in a
 % given eigenspace by translating by the F's and E's, but neither of those are