Quasitilted algebras admit a preprojective component

Quasitilted algebras are generalizations of tilted algebras. As a main result we show here that the Auslander-Reiten quiver of such an algebra has a preprojective component Let k be an algebraically closed field and A be a finite-dimensional k-algebra. We denote by mod A the category of finitely generated left A-modules. By 1A we denote the Auslander-Reiten quiver of A. Recall that the vertices of 1A correspond to the isomorphism classes of indecomposable finitely generated A-modules. The number of arrows from an indecomposable A-module X to an indecomposable Amodule Y is the dimension of the k-vector space rad(X,Y)/rad2(X,Y), where rad(-, -) denotes the Jacobson radical of mod A. We denote by TX = DTr X the Auslander-Reiten translate of the indecomposable A-module X. This is defined for each indecomposable module, and in case X is non-projective the translate TX will be indecomposable and non-injective. Dually there is defined T-X = TrDX. A connected component P of 1A is called a preprojective component if P does not contain an oriented cycle and each X c P is of the form T-rp for some r c N and an indecomposable projective A-module P. For details see [ARS]. The existence of preprojective components has been established for various classes of algebras such as tilted algebras [St] or algebras satisfying the separation condition [B]. It is well known that tilted algebras may have several preprojective components. One of the important features of an indecomposable module X lying in a preprojective component is that X is homologically trivial, i.e. ExtX(X, X) = 0 for i > 0 and EndAX = k and that its isomorphism class is uniquely determined by the composition factors. Also these modules can be constructed easily by the socalled knitting procedure. Quasitilted algebras have been introduced and investigated in [HRS2]. Recall that a finite-dimensional k-algebra A is called a quasitilted algebra if there exist a hereditary abelian k-category XH and a tilting object T c XH such that A = End'HT. In this article we will not work with this definition but rather with the homological characterization established in [HRS2]. We will use the following notation. For Received by the editors March 31, 1995 and, in revised form, November 10, 1995. 1991 Mathematics Subject Classification. Primary 16G10, 16E10.