Exceptional objects in hereditary categories

Ringel CM. Exceptional objects in hereditary categories. In: Roggenkamp KW, Stefanescu M, eds. Analele Stiintifice ale Universitatii Ovidius Constanta. Proceedings: Representation Theory of Groups, Algebras, and Orders. September 25 - October 6, 1995, Constanta. Seria Matematica. Vol 4. Faculty of Mathematics and Computer Science, Ovidius University, Constanta, Romania; 1996: 150-158.Let k be a field and A a finite dimensional k-category which is a hereditary length category. We are going to show that the support algebra of any object of A without self-extension is a finite dimensional k-algebra. An object in A is said to be exceptional provided it is indecomposable and has no self-extensions. For an algebraically closed field k, Schofield has exhibited an algorithm for obtaining all exceptional objects starting from the simple ones. We will present a proof which works for arbitrary fields k