The braid group action on the set of exceptional sequences of a hereditary artin algebra

Ringel CM. The braid group action on the set of exceptional sequences of a hereditary artin algebra. In: Göbel R, ed. Abelian group theory and related topics: Conference on Abelian Groups, August 1 - 7, 1993, Oberwolfach, Germany. Contemporary mathematics. Vol 171. Providence, RI: American Mathematical Soc.; 1994: 339-352.Let A be a hereditary artin algebra with s = s(A) simple modules. The indecomposable A-modules without self-extensions are of great importance, they may be called complete exceptional sequences. Certain sequences (X1,..., X3) consisting of exceptional modules will be called complete exceptional sequences. Crawley-Boevey has pointed out that the braid group on s-1 generators acts naturally on the set of complete exceptional sequences. In case A is finite-dimensional over an algebraically closed field, he has shown that this action is transitive, using a recnet result by Schofield. We are going to present a direct proof which is valid for arbitrary hereditary artin algebras. It follows that the endomorphism rings of exceptional modules are just those rings which occur as endomorphism rings of the simple modules. Also, we will exhibit the relationship between complete exceptional sequences and tilting modules