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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Outline}
  \tableofcontents
\end{frame}

\section{Examples of computations}

\begin{frame}[fragile]{$\Omega$-periodic modules}
$\Lambda$ -- finite dimensional algebra\medskip

\textbf{Recall:} A module $M$ is \emph{$\Omega$-periodic} if
$\Omega^n_\Lambda(M)\simeq M$ for some positive integer $n$.\medskip 

In some situations the period can indicate the degree of generators in
the Hochschild cohomology ring, $\oplus_{i\geqslant
  0}\Ext^i_{\Lambda^\env}(\Lambda, \Lambda)$. 
\end{frame}

\begin{frame}[fragile]{Periodic algebras}

$\Lambda$ -- finite dimensional $k$-algebra\medskip

$\Lambda^\env = \Lambda^\op\otimes_k \Lambda$ -- enveloping
algebra\medskip

\textbf{Recall:} $\Lambda$ is a \emph{periodic algebra} if $\Lambda$ is a
$\Omega$-periodic module, that is $\Omega^n_{\Lambda^\env}(\Lambda)
\simeq \Lambda$ as $\Lambda^\env$-modules.\medskip

\textbf{Facts:}
\begin{itemize}
\item $\Lambda$ is a selfinjective algebra.
\item All modules are $\Omega$-periodic.
\item The Hochschild cohomology module nilpotent elements is
  isomorphic to $k[x]$, where the degree of $x$ is the period.
\end{itemize}
\end{frame}

\begin{frame}[fragile]{Finding quivers}

\textbf{Recall:} $\Lambda = kQ/I$ with $I$ admissible is a basic algebra, that is, 
\[\Lambda = \oplus_{i=1}^t P_i\]
with $P_i$ indecomposable, then $P_i\not\simeq P_j$ for $i\neq j$. 

\textbf{Facts:} 
\begin{itemize}
\item $\Lambda$ -- finite dimensional algebra.
\item $\rad \Lambda = \langle \textrm{arrows}\rangle/I$.
\item $\Lambda/\rad\Lambda \simeq \frac{kQ/I}{\langle
    \textrm{arrows}\rangle/I} \simeq \textrm{linear span of vertices}$.
\item \begin{align}
    \rad\Lambda/\rad^2\Lambda & \simeq \frac{\langle
    \textrm{arrows}\rangle}{I}/\frac{\langle
    \textrm{arrows}\rangle^2}{I} \notag\\
    & \simeq \frac{\langle
    \textrm{arrows}\rangle}{\langle \textrm{arrows}\rangle^2}\notag\\
   & \simeq \textrm{linear span of arrows}\notag
\end{align}
\end{itemize}
\end{frame}

\begin{frame}[fragile]{Finding quivers}
$\Lambda$ -- finite dimensional algebra with $\Lambda/\rad\Lambda
\simeq k^n$ for some $n$.\medskip 

Algorithm:
\begin{enumerate}[\rm(1)]
\item Lift a complete set of orthogonal idempotents from
  $\Lambda/\rad\Lambda$ to a complete set of orthogonal idempotents in
  $\Lambda$, say $\{e_i\}_{i=1}^n$ -- the vertices. 
\item Compute $e_i\rad\Lambda/\rad^2\Lambda e_j$, find a basis and
  lift back to $e_i\rad\Lambda e_j$ -- the arrows from vertex $i$ to
  vertex $j$.   
\item Construct a quiver $Q$ from this and a homomorphism
  $\varphi\colon kQ \to \Lambda$. 
\item Find the kernel of $\varphi$. 
\end{enumerate}
\end{frame}

\begin{frame}[fragile]{Trivial extensions}
$\Lambda$ -- finite dimensional algebra\medskip

$T(\Lambda) = \Lambda\oplus D(\Lambda)$ -- \emph{trivial extension},

\[ (\lambda, f)\cdot (\lambda',f') = (\lambda\lambda', \lambda f' +
f\lambda)\]

\begin{itemize}
\item $\rad T(\Lambda) = \rad\Lambda \oplus D(\Lambda)$. 
\item $\rad^2 T(\Lambda) = \rad^2\Lambda \oplus D(\Lambda)\rad\Lambda
  + \rad\Lambda D(\Lambda)$. 
\item $\frac{\rad T(\Lambda)}{\rad^2 T(\Lambda)} \simeq
\frac{\rad\Lambda}{\rad^2\Lambda}\oplus \frac{D(\Lambda)}{D(\Lambda)\rad\Lambda
  + \rad\Lambda D(\Lambda)}$
\item $T(\Lambda)$ is a symmetric algebra, $\Gamma\simeq D(\Gamma)$ as
bimodules.
\end{itemize} 
\end{frame}

\section{Tilting modules}

\begin{frame}[fragile]{AR-theory}
Recall that a short exact sequence 
\[0\to A\xrightarrow{f} B\xrightarrow{g} C\to 0\]
is \emph{almost split exact} if it is not split exact and 
\begin{enumerate}[\rm(i)]
\item for any not splittable epimorphism $t\colon X\to C$ there is a
  homomorphism $t'\colon X\to B$ such that $gt' = t$, 
\item for any not splittable monomorphism $s\colon A\to Y$ there is a
  homomorphism $s'\colon A\to Y$ such that $s'f = s$.
\end{enumerate}
\end{frame}

\begin{frame}[fragile]{AR-theory}
Facts:
\begin{itemize}
\item $C$ and $A$ are indecomposable modules. 
\item $A\simeq D\Tr C$ and $C\simeq \Tr D(A)$.  
\item For any indecomposable non-projective module $C$ and
for any indecomposable non-injective module $A$, there is an almost
split sequence ending in $C$ and starting in $A$.  
\item An almost split sequence is a generator of the socle of
  $\Ext^1_\Lambda(C,D\Tr(C))$ as an $\End_\Lambda(C)$-module. 
\end{itemize}
\end{frame}

\begin{frame}[fragile]{APR-tilting}
$\Lambda=kQ$ -- hereditary, $Q$ no oriented cycle and connected.\medskip 

$S$ simple projetive module (and not injective)

$\Lambda$ is a classical tilting module $T$: 
\begin{itemize}
\item $\pd_\Lambda T \geqslant 1$, 
\item $\Ext^1_\Lambda(T,T) = (0)$, 
\item the number of indecomposable non-isomorphic summands in $T$ is
  equal to the number of isomorphism class of simple modules of
  $\Lambda$. 
\end{itemize}

$\Lambda = P \oplus S \longrightarrow T = P\oplus \Tr D(S)$ -- APR-tilting







\end{frame}
\end{document}
