diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -397,56 +397,56 @@ basic}. In fact, all we need to know is\dots
induces long exact sequences
\begin{center}
\begin{tikzcd}
- 0 \rar &
+ 0 \rar &
\operatorname{Hom}_{\mathfrak{g}}(L', N)
- \rar[swap]{f \circ -}\ar[draw=none]{d}[name=X, anchor=center]{} &
- \operatorname{Hom}_{\mathfrak{g}}(L', M) \rar[swap]{g \circ -} &
+ \rar{f \circ -}\ar[draw=none]{d}[name=X, anchor=center]{} &
+ \operatorname{Hom}_{\mathfrak{g}}(L', M) \rar{g \circ -} &
\operatorname{Hom}_{\mathfrak{g}}(L', L)
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (X.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
- -- (\tikztotarget)}]{dll}[at end]{} \\ &
+ -- (\tikztotarget)}]{dll}[at end]{} \\ &
\operatorname{Ext}^1(L', N)
- \rar\ar[draw=none]{d}[name=Y, anchor=center]{} &
- \operatorname{Ext}^1(L', M) \rar &
+ \rar\ar[draw=none]{d}[name=Y, anchor=center]{} &
+ \operatorname{Ext}^1(L', M) \rar &
\operatorname{Ext}^1(L', L)
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (Y.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
- -- (\tikztotarget)}]{dll}[at end]{} \\ &
- \operatorname{Ext}^2(L', N) \rar &
- \operatorname{Ext}^2(L', M) \rar &
- \operatorname{Ext}^2(L', L) \rar[dashed] &
+ -- (\tikztotarget)}]{dll}[at end]{} \\ &
+ \operatorname{Ext}^2(L', N) \rar &
+ \operatorname{Ext}^2(L', M) \rar &
+ \operatorname{Ext}^2(L', L) \rar[dashed] &
\cdots
\end{tikzcd}
\end{center}
and
\begin{center}
\begin{tikzcd}
- 0 \rar &
+ 0 \rar &
\operatorname{Hom}_{\mathfrak{g}}(L, L')
- \rar[swap]{- \circ g}\ar[draw=none]{d}[name=X, anchor=center]{} &
- \operatorname{Hom}_{\mathfrak{g}}(M, L') \rar[swap]{- \circ f} &
+ \rar{- \circ g}\ar[draw=none]{d}[name=X, anchor=center]{} &
+ \operatorname{Hom}_{\mathfrak{g}}(M, L') \rar{- \circ f} &
\operatorname{Hom}_{\mathfrak{g}}(N, L')
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (X.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
- -- (\tikztotarget)}]{dll}[at end]{} \\ &
+ -- (\tikztotarget)}]{dll}[at end]{} \\ &
\operatorname{Ext}^1(L, L')
- \rar\ar[draw=none]{d}[name=Y, anchor=center]{} &
- \operatorname{Ext}^1(M, L') \rar &
+ \rar\ar[draw=none]{d}[name=Y, anchor=center]{} &
+ \operatorname{Ext}^1(M, L') \rar &
\operatorname{Ext}^1(N, L')
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (Y.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
- -- (\tikztotarget)}]{dll}[at end]{} \\ &
- \operatorname{Ext}^2(L, L') \rar &
- \operatorname{Ext}^2(M, L') \rar &
- \operatorname{Ext}^2(N, L') \rar[dashed] &
+ -- (\tikztotarget)}]{dll}[at end]{} \\ &
+ \operatorname{Ext}^2(L, L') \rar &
+ \operatorname{Ext}^2(M, L') \rar &
+ \operatorname{Ext}^2(N, L') \rar[dashed] &
\cdots
\end{tikzcd}
\end{center}
@@ -519,9 +519,9 @@ This implies\dots
\begin{center}
\begin{tikzcd}
0 \rar &
- N^{\mathfrak{g}} \rar[swap]{f}
+ N^{\mathfrak{g}} \rar{f}
\ar[draw=none]{d}[name=X, anchor=center]{} &
- M^{\mathfrak{g}} \rar[swap]{g} &
+ M^{\mathfrak{g}} \rar{g} &
L^{\mathfrak{g}}
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
@@ -848,19 +848,19 @@ We are now finally ready to prove\dots
long exact sequence
\begin{center}
\begin{tikzcd}
- 0 \rar &
- \operatorname{Hom}(L, N)^{\mathfrak{g}} \rar[swap]{f \circ -}
- \ar[draw=none]{d}[name=X, anchor=center]{} &
- \operatorname{Hom}(L, M)^{\mathfrak{g}} \rar[swap]{g \circ -} &
+ 0 \rar &
+ \operatorname{Hom}(L, N)^{\mathfrak{g}} \rar{f \circ -}
+ \ar[draw=none]{d}[name=X, anchor=center]{} &
+ \operatorname{Hom}(L, M)^{\mathfrak{g}} \rar{g \circ -} &
\operatorname{Hom}(L, L)^{\mathfrak{g}}
\ar[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (X.center) \tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
- -- (\tikztotarget)}]{dll}[at end]{} \\ &
- H^1(\mathfrak{g}, \operatorname{Hom}(L, N)) \rar &
- H^1(\mathfrak{g}, \operatorname{Hom}(L, M)) \rar &
- H^1(\mathfrak{g}, \operatorname{Hom}(L, L)) \rar[dashed] &
+ -- (\tikztotarget)}]{dll}[at end]{} \\ &
+ H^1(\mathfrak{g}, \operatorname{Hom}(L, N)) \rar &
+ H^1(\mathfrak{g}, \operatorname{Hom}(L, M)) \rar &
+ H^1(\mathfrak{g}, \operatorname{Hom}(L, L)) \rar[dashed] &
\cdots
\end{tikzcd}
\end{center}