Exceptional objects in hereditary categories

Claus Michael Ringel

Abstract. 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 close 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.