Auslander Reiten Components Research Articles

Filtration-Closed Auslander-Reiten Components for Wild Hereditary Algebras

Let H=kQ be a finite-dimensional connected wild hereditary path algebra, over some field k. Denote by H-reg the category of finite-dimensional regular H-modules, that is, the category of modules M with τ H − m ( τ H m M ) ≅ M for all integers m, where τH denotes the Auslander–Reiten translation. Call a filtration ( ∗ ) M = M 0 ⊃ M 1 ⊃ … ⊃ M r ⊃ M r + 1 = 0 of a regular H-module M a regular filtration if all subquotients Mi/Mi+1 are regular. Call a regular filtration (*) a regular composition series if it is strictly decreasing and has no proper refinement. A regular component C in the Auslander–Reiten quiver Γ (H) of H-mod is called filtration closed if, for each M∈ add C, the additive closure of C, and each regular filtration (*) of M, all the subquotients Mi/Mi+1 are also in add C. We show that most wild hereditary algebras have filtration-closed Auslander–Reiten components. Moreover, we deduce from this that there are also almost serial components, that is regular components C, such that any indecomposable X∈C has a unique regular composition series. This composition series coincides with the Auslander–Reiten filtration of X, given by the maximal chain of irreducible monos ending at X. 1991 Mathematics Subject Classification: 16G70, 16G20, 16G60, 16E30.

On the Auslander–Reiten quiver of a [formula omitted]-hereditary artin algebra

We investigate the Auslander–Reiten quiver of a P 1 -hereditary artin algebra Λ by relating it to the Auslander–Reiten quiver of a suitable tilted algebra. We then obtain that the Auslander–Reiten components of Λ are postprojective, or preinjective, or quasi-serial, or obtained from a quasi-serial translation quiver by coray insertions, or they can be interpreted as glueings of a finite number of components which are either postprojective or preinjective.

On the stable module category of a self-injective algebra

Let A be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable Amodules which belong to Auslander-Reiten components of the form ZA or ZAo / (rk). The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type. We are interested in homological properties of modules for self-injective algebras. An important invariant of a finite-dimensional algebra is its stable Auslander-Reiten quiver; and during the past years, it has played a crucial role for classification problems of self-injective algebras of finite or tame representation type. We call a stable component quasi-serial, if it is of the form ZAoo or ZAoo/(rk). If A is any tame algebra, then most of its Auslander-Reiten components are homogeneous tubes, by [CB], hence are quasi-serial. Moreover, it seems that for self-injective algebras of wild type most components of the stable Auslander-Reiten quiver are quasi-serial. For a block of a group algebra which is of wild representation type, all components are of this form (see [E2]). Therefore, it is important to understand homological properties of quasi-serial components. If A is a finite-dimensional self-injective algebra and X, Y are A-modules, then we denote the stable homomorphisms from X to Y by Hom(X, Y). Following the terminology in [RI], if M is a module in a quasi-serial stable component C, then the quasi-length of M is the number of the row to which M belongs. The module M is quasi-simple if it has quasi-length one, that is, if it lies at the end of the component. The first two chapters contain basic facts on dimensions of spaces of stable homomorphisms; this is intended to provide tools which may be of more general use. In Chapter 3 we prove the following general results. Suppose f is an equivalence of the stable module category of A, consider stable homomorphisms Hom(M, fM) for modules M in a quasi-serial component C. We prove that the dimension of Hom(M, fM) is weakly increasing as a function of the quasi-length of M, except possibly when TQ-1 f T on C (see 3.3). Moreover, we study the cases when f = IS and f = QS in more detail. We show that if C is a tube, then the dimensions of Hom(M, rSM) for M e C are unbounded (3.5). In general, if C is a component of the form ZA,, which is not fixed by Q and the dimensions of Hom(M, TsM) are bounded, then Hom(X, r-mX) 0 Received by the editors June 4, 1996 and, in revised form, October 2, 1997. 2000 Mathematics Subject Classification. Primary 18G25; Secondary 16G70, 20C20.

Roots of Nakayama and Auslander-Reiten translations

N) Abstract. We discuss the roots of the Nakayama and Auslander-Reiten translations in the derived category of coherent sheaves over a weighted projective line. As an appli- cation we derive some new results on the structure of selfinjective algebras of canonical type. Throughout this paper K will denote a fixed algebraically closed field. We work in the derived category D b (X) of the category cohX of coherent sheaves on a weighted projective line X over K. We investigate whether, for a positive integer d, one of the automorphisms �T 2 ; ├T 2 ; �; that is, the Nakayama translation, a twisted Nakayama translation or the Auslander-Reiten translation, respectively, has a dth root in the automor- phism group of D b (X). Here, % denotes a rigid automorphism, that is, an automorphism of cohX—identified with a member of Aut(D b (X))—which preserves all Auslander-Reiten components and also the slope of indecom- posable objects; further, T denotes the translation shift in the derived cat- egory D b (X). Let Pic0 X denote the torsion group of the Picard group of X, and let AutX denote the automorphism group of X, identified with the group of all isomorphism classes of selfequivalences of the category cohX fixing the structure sheaf. It then follows from (9) that the rigid automorphisms form a subgroup of Pic0 Xo AutX, and, moreover, this group is finite if X has at least three exceptional points. Throughout the paper, by an automorphism we mean the isomorphism class of a selfequivalence of K-categories. When applied to a finite-dimen- sional basic K-algebra A, this means to identify automorphisms that differ by an inner automorphism. In particular, we say that an automorphism of A is non-trivial if it is not inner.

Wild canonical algebras

Canonical algebras ￿=￿(p; ￿); depending on a weight sequence p=(p1; : : : ; pt) of positive integers, and a parameter sequence ￿ = (￿3; : : : ; ￿t) of pairwise distinct non-zero elements from the base ￿eld k; were introduced and studied in [23]. The ￿nite dimensional representation theory mod(￿) is completely controlled by the category coh(X) of coherent sheaves on a non-singular weighted projective line X = X(p; ￿); since the derived categories D(mod(￿)) and D(coh(X)) are equivalent as triangulated categories [9]. The curve X attached to ￿ has (virtual) genus gX = 1 + 2((t − 2)p − ￿ t i=1 p=pi); where p=l:c:m:(p1; : : : ; pt): The complexity of the classi￿cation problem for coh(X); and hence for mod(￿); is essentially determined by gX: For gXi1; the algebra ￿ is concealed of extended Dynkin type; accordingly the classi￿cation problem for coh(X) and mod(￿) is equivalent to the classi￿cation of indecomposable modules over a tame hereditary algebra (cf. [18], [9]) or, according to [10], to the classi￿cation problem for Cohen–Macaulay modules over a simple surface singularity. For gX = 1; the algebra ￿ is of tubular type, its representation theory is known from [23], while the classi￿cation problem for coh(X) [9, 20] relates to Atiyah’s classi￿cation [1] of vector bundles over an elliptic curve. Very little was known on mod(￿) or coh(X) if gX?1. In this case, ￿= ￿0[R] is the one-point extension of a wild hereditary algebra ￿0; accordingly the category mod(￿0) embeds into coh(X) as a full, exact and extension-closed subcategory, implying that the classi￿cation problems for coh(X) and mod(￿) are wild. From now on we assume gX ? 1: Combining the sheaf theory [9] with the spectral investigation of wild hereditary algebras [6], we obtain a fairly complete picture on the structure of Auslander–Reiten components for mod(￿); the asymptotic behaviour of the Auslander–Reiten translations ￿￿ and

On Auslander–Reiten components for quasitilted algebras

An artin algebra A over a commutative artin ring R is called quasitilted if gl.dim A ≤ 2 and for each indecomposable finitely generated A-module M we have pd M ≤ 1 or id M ≤ 1. In [11] several characterizations of quasitilted algebras

The existence of bounded infinite 𝐷Tr-orbits

We construct an indecomposable module over a symmetric algebra whose DTr-ovbil is infinite and bounded. This yields a counterexample to a conjecture which states that the number of modules in an Auslander-Reiten component having the same length is finite.

Regular Auslander-Reiten components containing directing modules

We describe completely all regular Auslander-Reiten components of artin algebras containing directing modules. We prove also that the Auslander-Reiten quiver of an artin algebra admits at most finitely many DTr-orbits containing directing modules.

On Auslander-Reiten Components of Blocks and Self-Injective Biserial Algebras

We investigate the existence of Auslander-Reiten components of Euclidean type for special biserial self-injective algebras and for blocks of group algebras. In particular we obtain a complete description of stable Auslander-Reiten quivers for the tame self-injective algebras considered here.

On Auslander-Reiten components of blocks and self-injective biserial algebras

We investigate the existence of Auslander-Reiten components of Euclidean type for special biserial self-injective algebras and for blocks of group algebras. In particular we obtain a complete description of stable Auslander-Reiten quivers for the tame self-injective algebras considered here.