Extend central charge to derived category

main.tex

@@ -30,6 +30,7 @@
 \newcommand\realpart{\mathfrak{Re}}
 \newcommand\imagpart{\mathfrak{Im}}
 \newcommand\bigO{\mathcal{O}}
+\newcommand\cohom{\mathcal{H}}
 
 \begin{document}
 
@@ -90,6 +91,36 @@
 
 \end{frame}
 
+\begin{frame}{Extending Central Charge to $D^b(X)$}
+	\[
+		E^\bullet = [\cdots \to * \to * \to * \to \cdots] \in D^b(X)
+	\]
+	\begin{align*}
+		\rank(E^\bullet) &= \sum (-1)^i \rank(\cohom^i(E)) \\
+		\degree(E^\bullet) &= \sum (-1)^i \degree(\cohom^i(E))
+	\end{align*}
+
+	\vfill
+
+	\begin{columns}[t,onlytextwidth]
+		\begin{column}{.5\linewidth}
+			In particular, for shifts:
+			\begin{itemize}
+				\item $\rank(E[1]) = - \rank(E)$
+				\item $\degree(E[1]) = - \degree(E)$
+			\end{itemize}
+		\end{column}
+		\begin{column}{.49\linewidth}
+			For $\centralcharge$:
+			\begin{itemize}
+				\item $\centralcharge(E[1]) = - \centralcharge(E)$
+				\item $\centralcharge(E[2]) = \centralcharge(E)$
+			\end{itemize}
+		\end{column}
+	\end{columns}
+
+\end{frame}
+
 \begin{frame}{title}
 \begin{columns}[t,onlytextwidth] % align columns
 	\begin{column}{.33\linewidth}%
@@ -116,7 +147,8 @@
 	\begin{itemize}
 		\item $\centralcharge(\bigO_X) = \centralcharge(\bigO_X[2])$
 		\item But $\phi(\bigO_X) \not= \phi(\bigO_X[2])$
-		\item Stability decided among $E \in D^b(X) \colon \phi(E) \in (0, \pi]$ \\
+		\item Stability decided among $E \in D^b(X) \colon \phi(E) \in (0, \pi] =
+			\coh(X)$ \\
 			($\phi$ only partial "function")
 	\end{itemize}
 \end{frame}