lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -496,15 +496,15 @@
 \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
-  ring homomorphism \(R \to S^{-1} R\), and enjoying the universal property that
-  each ring homomorphism \(f : R \to T\) such that \(f(s)\) is invertible for
-  all \(s \in S\) can be uniquely extended to a ring homomorphism \(S^{-1} R
-  \to T\). \(S^{-1} R\) is called \emph{the localization of \(R\) by \(S\)},
+  ring homomorphism \(R \to S^{-1} R\), and enjoying the universal property
+  that each ring homomorphism \(f : R \to T\) such that \(f(s)\) is invertible
+  for all \(s \in S\) can be uniquely extended to a ring homomorphism \(S^{-1}
+  R \to T\). \(S^{-1} R\) is called \emph{the localization of \(R\) by \(S\)},
   and the map \(R \to S^{-1} R\) is called \emph{the localization map}.
   \begin{center}
     \begin{tikzcd}
-      S^{-1} R \rar[dotted] & T \\
-      R \arrow{u} \arrow[swap]{ur}{f} &
+      S^{-1} R \arrow[dotted]{rd} & \\
+      R \arrow{u} \arrow[swap]{r}{f} & T
     \end{tikzcd}
   \end{center}
 \end{theorem}