On Auslander–Reiten components for quasitilted algebras

An artin algebra A over a commutative artin ring R is called quasitilted if gl.dimA ≤ 2 and for each indecomposable finitely generated A-module M we have pdM ≤ 1 or idM ≤ 1. In [11] several characterizations of quasitilted algebras were proven. We investigate the structure and homological properties of connected components in the Auslander–Reiten quiver ΓA of a quasitilted algebra A. Let A be an artin algebra over a commutative artin ring R, that is, A is an R-algebra which is finitely generated as an R-module. Denote by indA the category of indecomposable finitely generated right A-modules, by ΓA the Auslander–Reiten quiver of A, and by τA the Auslander–Reiten translation in ΓA. Following [10], the algebra A is called tilted if there exists a hereditary artin algebra H and a tilting H-module T such that A = EndH(T ). Recall that a finitely generated H-module T is called tilting if ExtH(T, T ) = 0 and there is an exact sequence 0 → HH → T0 → T1 → 0 with T0 and T1 in the additive category addT , given by T . The representation theory of tilted algebras is fairly well understood. In particular, we know the shape of all connected components of the Auslander– Reiten quivers of tilted algebras (see [8], [12], [13], [17]–[20], [27]). It is known that a tilted algebra A is of global dimension at most 2 and no module in indA has both projective and injective dimension equal to 2. However, these properties do not characterize the tilted algebras. Happel, Reiten and Smalø have shown in [11] that they characterize the class of quasitilted algebras which are the artin algebras of the form A = End(T ), where T is a tilting object in a hereditary abelian R-category H. Besides the tilted algebras, important classes of quasitilted algebras are provided by tubular algebras [19], canonical algebras [14], [19], [21], algebras with separating tubular families of modules [15], [25], and semiregular 1991 Mathematics Subject Classification: 16G10, 16G70, 18G20.