Hereditary categories with tilting object

Let k be an algebraically closed field and H a connected abelian k-category which is hereditary, that is Ext2( , ) vanishes on H. Assume also that Hom(X, Y ) and Ext1(X, Y ) are finite dimensional vector spaces over k for all X and Y in H. We consider such hereditary categories H which in addition have a tilting object T , that is, an object T such that {X; Ext1(T, X) = 0} = FacT , the factors of finite direct sums of copies of T . Hereditary categories H with a tilting object T are of special interest in connection with the construction of the class of algebras called quasitilted algbras, which was introduced in [HRS1]. They are by definition the algebras of the form EndH(T )op. An important property is that H and EndH(T )op have equivalent bounded derived categories. The main examples of such hereditary categories are the category modH of finitely generated modules over a finite dimensional hereditary k-algebra H and the category cohX of coherent sheaves on a weighted projective line in the sense of [GL1]. There are also others derived equivalent to them. Because of the simple description of the corresponding bounded derived category Db(H), it is possible to give a description of those in the same derived equivalence class. (See [LS] [H2]). It follows from [HRe] that they automatically have a tilting object. The quasitilted algebras provide common generalization for the tilted and the canonical finite dimensional algebras. Note that the tilted algebras are those coming from modH using an arbitrary