diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -266,7 +266,7 @@
\end{enumerate}
\end{theorem}
-\section{Existance of Coherent Extensions}
+\section{Localizations \& 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
@@ -274,30 +274,86 @@
% If the F's were invertible this problem wouldn't exist, so we might as well
% invert them by force!
-% TODO: Add the results on Ore's localization
+\begin{definition}
+ Let \(R\) be a ring. A subset \(S \subset R\) is called \emph{multiplicative}
+ if \(s \cdot t \in S\) for all \(s, t \in S\) and \(0 \notin S\).
+ A multiplicative subset \(S\) is said to satisfy \emph{Ore's localization
+ condition} if for each \(r \in R\), \(s \in S\) there exists \(u_1, u_2 \in
+ R\) and \(t_1, t_2 \in S\) such that \(s^{-1} r = u_1 t_1^{-1}\) and \(r
+ s^{-1} = t_2^{-1} u_2\).
+\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
+ 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} &
+ \end{tikzcd}
+ \end{center}
+\end{theorem}
+
+% TODO: Cite the discussion of goodearl-warfield, chap 6, on how to derive the
+% localization condition
+% TODO: In general checking that a set satisfies Ore's condition can be tricky,
+% but there is an easyer condition given by the lemma
+
+\begin{lemma}
+ Let \(S \subset R\) be a multiplicative subset generated by locally
+ \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\) such
+ that for each \(r \in R\) there \([s, [s, \cdots [s, r]]\cdots] = 0\) for
+ sufficiently many applications of the commutator. Then \(S\) satisfies Ore's
+ localization condition.
+\end{lemma}
+
+\begin{definition}
+ Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+ condition and \(M\) be a \(R\)-module. The \(S^{-1} R\)-module \(S^{-1} M =
+ S^{-1} R \otimes_R M\) is called \emph{the localization of \(M\) by \(S\)},
+ and the homomorphism of \(R\)-modules
+ \begin{align*}
+ M & \to S^{-1} M \\
+ m & \mapsto 1 \otimes m
+ \end{align*}
+ is called \emph{the localization map of \(M\)}.
+\end{definition}
+
+% TODO: Point out the the localization map is in not injective in general
+\begin{lemma}
+ Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+ condition and \(M\) be a \(R\)-module. If \(S\) acts injectively in \(M\)
+ then the localization map \(M \to S^{-1} M\) is injective. In particular, if
+ \(S\) has no left zero divisors then \(R\) is a subring of \(S^{-1} R\).
+\end{lemma}
+
+% TODO: Point out that S^-1 M can be seen as a R-module, where R acts via the
+% localization map
+% TODO: Point out that each element of the localization has the form s^-1 r
% TODO: Define what a set commuting roots is
\begin{lemma}\label{thm:nice-basis-for-inversion}
Let \(V\) be an irreducible infinite-dimensional admissible
\(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
- \beta_n\}\) of \(Q\) consisting a commuting roots and such that the elements
- \(F_{\beta_i}\) all act injectively on \(V\).
+ \beta_n\}\) of \(\Delta\) consisting a commuting roots and such that the
+ elements \(F_{\beta_i}\) all act injectively on \(V\).
\end{lemma}
\begin{corollary}
Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
- \((F_\beta : \beta \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
- the multiplicative subset \((F_\beta)_{\beta \in \Sigma}\) generated by the
- \(F_\beta\)'s.
- The \(K\)-algebra \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta
- \in \Sigma}^{-1} \mathcal{U}(\mathfrak{g})\) is well defined. Moreover, if we
- denote by \(\Sigma^{-1} V\) the localization of \(V\) by \((F_\beta)_{\beta
- \in \Sigma}\), the localization map
- \begin{align*}
- V & \to \Sigma^{-1} V \\
- v & \mapsto 1 \otimes v
- \end{align*}
- is injective.
+ \((F_\beta)_{\beta \in \Sigma} \subset \mathcal{U}(\mathfrak{g})\) be the
+ multiplicative subset \((F_\beta)_{\beta \in \Sigma}\) generated by the
+ \(F_\beta\)'s. The \(K\)-algebra \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) =
+ (F_\beta)_{\beta \in \Sigma}^{-1} \mathcal{U}(\mathfrak{g})\) is well
+ defined. Moreover, if we denote by \(\Sigma^{-1} V\) the localization of
+ \(V\) by \((F_\beta)_{\beta \in \Sigma}\), the localization map \(V \to
+ \Sigma^{-1} V\) is injective.
\end{corollary}
% TODO: Fix V and Sigma beforehand
@@ -338,8 +394,6 @@
= F_\beta^{-1} (\lambda + \beta)(H) \cdot v
\]
- % TODO: Remark beforehand that any element of the localization of V may be
- % written as an element of v tensored by an element of the form 1/s
From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(\Sigma^{-1}
V_{\lambda \pm \beta}\) follows our first conclusion: since \(V\) is a weight
module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
@@ -365,8 +419,6 @@
\Sigma^{-1}_\lambda = d\).
\end{proof}
-% TODO: Remark that any module over the localization is a g-module if we
-% restrict it via the localization map, wich is injective in this case
\begin{proposition}\label{thm:nice-automorphisms-exist}
There is a family of automorphisms \(\{\theta_\lambda : \Sigma^{-1}
\mathcal{U}(\mathfrak{g}) \to \Sigma^{-1}