Finite cycles of indecomposable modules

Abstract

Abstract. We solve a long standing open problem concerning the structure of finite cycles in the category mod A of finitely generated modules over an arbitrary artin algebra A, that is, the chains of homomorphisms M 0 f 1 −−→ M 1 → · · · → M r−1 fr −−→ Mr = M 0 between indecomposable modules in mod A which do not belong to the infinite radical of mod A. In particular, we describe completely the structure of an arbitrary module category mod A whose all cycles are finite. The main structural results of the paper allow to derive several interesting combinatorial and homological properties of indecomposable modules lying on finite cycles. For example, we prove that for all but finitely many isomorphism classes of indecomposable modules M lying on finite cycles of a module category mod A the Euler characteristic of M is well defined and nonnegative. As an another application of these results we obtain a characterization of all cycle-finite module categories mod A having only a finite number of functorially finite torsion classes. Moreover, new types of examples illustrating the main results of the paper are presented. IntroductionThroughout the paper, by an algebra is meant an artin algebra over a fixed commutative artin ring K, which we shall assume (without loss of generality) to be basic and indecomposable. For an algebra A, we denote by mod A the category of finitely generated right A-modules and by ind A the full subcategory of mod A formed by the indecomposable modules. The Jacobson radical rad A of mod A is the ideal generated by all nonisomorphisms between modules in ind A, and the infinite radical rad An important combinatorial and homological invariant of the module category mod A of an algebra A is its Auslander-Reiten quiver Γ A . Recall that Γ A is a valued translation quiver whose vertices are the isomorphism classes {X} of modules X in ind A, the arrows correspond to irreducible homomorphisms between modules in ind A, and the translation is the Auslander-Reiten translation τ A = DTr.We shall not distinguish between a module in X in ind A and the corresponding vertex {X} of Γ A . If A is an algebra of finite representation type, then every nonzero nonisomorphism in ind A is a finite sum of composition of irreducible homomorphisms between modules in ind A, and hence we may recover mod A from the translation quiver Γ A . In general, Γ A describes only the quotient category mod A/rad ∞ A . Let A be an algebra and M a module in ind A. An important information concerning the structure of M is coded in the structure and properties of its support algebra Supp(M ) defined as follows. Consider a decomposition A = P M ⊕ Q M of A in mod A such that the simple summands of the semisimple module P M /radP M are exactly the simple composition factors of M . Then Supp(M ) = A/t A (M ), where t A (M ) is the ideal in A generated by the images of all homomorphisms from Q M to A in mod A.We note that M is an indecomposable module over Supp(M ). Clearly, we may realistically hope to describe the structure of Supp(M ) only ...

Cycles of modules and finite representation type

“…In general, in order to obtain information on a module M in ind A, it is natural to study properties of cycles in ind A containing M . Following [21], a module M in ind A is said to be cycle-finite if M is not directing and every cycle in ind A passing through M is finite. Further, following [1,2], by a cycle-finite algebra we mean an algebra A for which all cycles in ind A are finite.…”

“…We will apply the following result established in [24, Corollary 3.5] (whose proof uses heavily results of [21,22,42]).…”

Cycle-Finite Algebras with Finitely Many τ-Rigid Indecomposable Modules

“…We have the following structure theorem, which is a consequence of a more general result proved in [20,Theorem 1.7] (see also [18, (i) All but finitely many isomorphism classes of modules in ind A belong to p i=1 ind B i ; (ii) All but finitely many isomorphism classes of nondirecting modules in ind A belong to generalized multicoils of B 1 B p .…”

“…The study of cycle-finite algebras have attached much attention. We refer to [10], [18], [19], [20], [33], [38], [39] for some general results on cycle-finite algebras and their module categories.…”

Krull Dimension of Tame Generalized Multicoil Algebras

“…Moreover, recently the tame generalized multicoil algebras showed to be important in describing the structure of the module category ind of an arbitrary cycle-finite algebra (see [18,Theorems 7.1 and 7.2] or [19,Theorem 1.8]). We also refer to the article [24] for the Hochschild cohomology of generalized multicoil algebras.…”