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