On cycles of length three (original) (raw)

On the composite of three irreducible morphisms over string algebras

The São Paulo Journal of Mathematical Sciences, 2010

We characterize the representation-finite triangular string algebras having a path of irreducible morphisms of length three between pairwise non-isomorphic modules whose composite lies in the fourth power of the radical.

Generic decompositions and semi-invariants for string algebras

2011

In this paper we investigate the rings of semi-invariants for tame string algebras A(n) of non-polynomial growth. We are interested in dimension vectors of band modules. We use geometric technique related to the description of coordinate rings on varieties of complexes. The fascinating combinatorics emerges, showing that our rings of invariants are the rings of some toric varieties. We show that for n ≤ 6 the rings of semi-invariants are complete intersections but we show an example for n = 7 that this is not the case in general.

A note on indecomposable modules

Rendiconti del Circolo Matematico di Palermo, 1988

In this note we study rings having only a finite number of non isomorphic uniform modules with non zero socle. It is proved that a commutative ring with this property is a direct sum of a finite ring and a ring of finite representation type. In the non commutative case we show that most P.I. rings having only a finite number of non isomorphic modules with non zero socle are in fact artinlan.

On algebras of finite representation type

Transactions of The American Mathematical Society, 1969

Introduction. Since D. G. Higman proved that bounded representation type and finite representation type are equivalent for group algebras at prime characteristic, there has been a renewed interest in the Brauer-Thrall conjecture that bounded representation type implies finite representation type for arbitrary algebras. The main purpose of this paper is to present a new approach to this conjecture by showing the relevance (when the base field is algebraically closed) of questions concerning the structure of indecomposable modules of certain special types, namely, the stable (every maximal submodule is indecomposable), the costable (having the dual property), and the stable-costable (having both properties) indecomposable modules. The main tools are the Sandwich Lemma (1.2) which is proved using an old observation of É. Goursat, an observation of A. Heller, C. W. Curtis, and D. Zelinsky concerning quasifrobenius (QF) rings (Proposition 2.1), and a general interlacing technique similar to methods used by Jans, Tachikawa, and Colby for building up large indecomposable modules of finite length which has validity in any abelian category (Theorem 3.1).

On the composite of two irreducible morphisms in radical cube

Journal of Algebra, 2007

We study here when there are irreducible morphisms h : X → Y and h : Y → Z between indecomposable modules such that the composite h h is a non-zero morphism in 3 (X, Z). In particular, we characterize the representation-finite string algebras and the tilted algebras having such irreducible morphisms.

Indecomposable Non Uniserial Modules of Length Three

International Electronic Journal of Algebra, 2020

We investigate a particular class of indecomposable modules of length three, defined over a K-algebra, with a simple socle and two non isomorphic simple factor modules. These modules may have any projective dimension different from zero. On the other hand their composition factors may have any countable dimension as vector spaces over the underlying field K. Moreover their endomorphism rings are K-vector spaces of dimension ≤ 2.

The composite of irreducible morphisms in regular components

Colloquium Mathematicum, 2011

We study when the composite of n irreducible morphisms between modules in a regular component of the Auslander-Reiten quiver is non-zero and lies in the n + 1-th power of the radical of the module category. We prove that in this case such a composite belongs to ∞. We apply these results to characterize those string algebras having n irreducible morphisms between band modules such that their composite is a non-zero morphism in n+1 .

Constructing Big Indecomposable Modules

2013

Abstract. Let R be local Noetherian ring of depth at least two. We prove that there are indecomposable R-modules which are free on the punctured spectrum of constant, arbitrarily large, rank. 1. introduction A fruitful approach to study a commutative ring is to understand the category of its finitely generated modules, and in particular the indecomposable objects of such a category. Over zero dimensional rings it is feasible to understand all the

Semi-Invariants for Gentle String Algebras

Eprint Arxiv 1106 0774, 2011

In this article we give an algorithm for determining the generators and relations for the rings of semi-invariant functions on irreducible components of representation spaces for gentle string algebras. These rings of semi-invariants turn out to be semigroup rings to which we can associate a so-called matching graph. Under this association, generators for the semigroup can be seen by certain walks on this graph, and relations are given by certain configurations in the graph. This allows us to determine degree bounds for the generators and relations of these rings. We furthermore show that these bounds also hold for acyclic string algebras in general.