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\centerline{\Large \bf On hereditary algebras}
\medskip

\centerline{\bf Steffen K\"onig}
\bigskip

{\bf Hereditary algebras} form one of the most intensively studied classes
of finite dimensional algebras (see \cite{Ri} for the present state of
knowledge). They are important both inside representation theory and for
several applications. Our main result underlines this importance even more. 

\begin{Theo} Let $A$ be a finite dimensional algebra over a perfect field.
Then $A$ is hereditary.
\end{Theo}

The {\bf Proof} is based on a series of lemmas. 

\begin{Lem} An algebra of global dimension zero is semisimple. A semisimple
algebra has global dimension zero. 

\noindent
An algebra of global dimension zero or one is hereditary.
\end{Lem}

\begin{Lem} A quotient of a semisimple algebra is semisimple, hence of
global dimension zero. By dimension shift \cite{CE}, a subalgebra of a
semisimple algebra has global dimension zero or one.
\end{Lem}

\begin{Lem} A finite dimensional algebra is a subalgebra of a semisimple
algebra.
\end{Lem}

\noindent
This finishes the proof of the theorem. \fatbox
\bigskip

Based on extensive computations for group algebras of finite groups we
conjecture that even more is true.

\begin{Conj} An algebra over a field of characteristic zero is semisimple.
\end{Conj}

{\footnotesize \begin{thebibliography}{9}
\bibitem{CE} {{\sc H.Cartan and S.Eilenberg,} Homological algebra, Princeton
University Press (1956).}
\bibitem{Ri}{{\sc C.M.Ringel,} Towards a representation theory for Dynkin
diagrams. Talk at ICRA VIII (1996).}
\end{thebibliography}}
\bigskip

\small
\noindent
{\sc Steffen K\"onig,
Fakult\"at f\"ur Mathematik,
Universit\"at Bielefeld,
Postfach 10 01 31,
D--33501 Bielefeld (Germany)}

\noindent
{\it E--mail address:} koenig@mathematik.uni-bielefeld.de
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