Lectures on algebras

the purpose of this note is to give a fast introduction to some problems of homological and geometrical nature related to finitely dimensional representations of finitely generated, and especially, finitely dimensional algebras over a field. some of these results can also be extended to the situation where the field is not algebraically closed, and some of the results can even be extended to the situation where one is considering algebras over a commutative artin ring. for the results which hold true in the most general situation the proofs become most elegant since they depend on using length arguments only and thereby forgetting about the nature of a field altogether.