Indecomposable Λ-modules Research Articles

The radical of functorially finite subcategories

Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if radC∞ vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.

Cohen–Macaulay Auslander algebras of gentle algebras

For a gentle algebra , we obtain an explicit description of the quiver with relations for the Cohen–Macaulay Auslander algebra . As a consequence, the Cohen–Macaulay Auslander algebra is again a gentle algebra. We then study the link between the representation theory of and the one of Λ. Among others, it is proved that is representation-finite if and only if Λ is; if Q has no loop and each indecomposable Λ-module is uniquely determined by its dimension vector, then each indecomposable -module is uniquely determined by its dimension vector.

On centrally generically tame algebras over perfect fields

We show that the central generic tameness of a finite-dimensional algebra Λ over a (possibly finite) perfect field, is equivalent to its non-almost sharp wildness. In this case: we give, for each natural number d, parametrizations of the indecomposable Λ-modules with central endolength d, modulo finite scalar extensions, over rational algebras. Moreover, we show that the central generic tameness of Λ is equivalent to its semigeneric tameness, and that in this case, algebraic boundedness coincides with central finiteness for generic Λ-modules.

Static Subcategories of the Module Category of a Finite-Dimensional Hereditary Algebra

Let k be a field, Λ a finite-dimensional hereditary k-algebra, and modΛ the category of all finite-dimensional Λ-modules. We are going to characterize the representation type of Λ (tame or wild) in terms of the possible subcategories statM of all M-static modules, where M is an indecomposable Λ-module.

On Semigenerically Tame Algebras Over Perfect Fields

Given a semigenerically tame finite-dimensional algebra Λ over a (possibly finite) perfect field, we give, for each natural number d, parametrizations of the indecomposable Λ-modules with central endolength bounded by d, modulo finite scalar extensions, over polynomial algebras.

Generic modules of tame algebras over real closed fields

Given a generically tame finite-dimensional algebra Λ over a real closed field, we describe the relations between the infinite families of indecomposable Λ-modules with bounded dimension and the generic Λ-modules. These are similar to those occurring for the algebraically closed field case, but the parametrizations are obtained over five particular centrally bounded principal ideal domains, instead of over rational algebras.

On the dimensions of path algebras

1. IntroductionThe notion of representation dimension of an artin algebra Λ, denoted rep.dimΛ, hasbeen introduced by Auslander in his Queen Mary Notes in 1971. His hope was to provide“a reasonable way of measuring how far the category of finitely generated modules over anartin algebra is from being of finite representation type”. Recall that an artin algebra Λis called of finite representation type if, up to isomorphisms, there are only finitely manyfinitely generated indecomposable Λ-modules. Such algebras are called representation finite.The importance of this class of artin algebras for the whole representation theory of artinalgebras is well understood, see e.g. [Au], [Ga] and [Ta].Auslander proved that Λ is of finite representation type if and only if its representationdimension is at most two. But it turned out to be hard to compute the actual value ofthe representation dimension of a given algebra. Moreover, all the algebras that one couldcompute explicitly their representation dimensions, had representation dimension at mostthree. So a natural question arises: are there artin algebras of representation dimensiongreater than three. Of course such algebras are of infinite representation type. On the otherhand, another question was whether this invariant is always finite. These questions has beenanswered positively, at almost the same time. Iyama [I] in 2003 proved that it is alwaysfinite and Rouquier [Rq1] in 2005 showed that any natural number n, except 1, can occuras the representation dimension of some algebra. In fact, he showed that the representationdimension of the exterior algebra Λk

On generically tame algebras over perfect fields

Given a generically tame finite-dimensional algebra Λ over an infinite perfect field, we give, for each natural number d, parametrizations of the indecomposable Λ-modules with dimension d similar to those occurring for the algebraically closed field case. We parametrize over bounded principal ideal domains, instead of over rational algebras.

Indecomposables live in all smaller lengths

Let Λ be a finite-dimensional k-algebra with k algebraically closed. Bongartz has recently shown that the existence of an indecomposable Λ-module of length n > 1 implies that also indecomposable Λ-modules of length n − 1 exist. Using a slight modification of his arguments, we strengthen the assertion as follows: If there is an indecomposable module of length n, then there is also an accessible one. Here, the accessible modules are defined inductively, as follows. First, the simple modules are accessible. Second, a module of length n ⩾ 2 is accessible provided it is indecomposable and that there is a submodule or a factor module of length n − 1 which is accessible.

Skew group algebras of deformed preprojective algebras

Suppose that Q is a finite quiver and G ⊆ Aut ( Q ) is a finite group, k is an algebraic closed field whose characteristic does not divide the order of G. For any algebra Λ = k Q / I , where I is an arbitrary ideal of path algebra kQ, we give all the indecomposable ΛG-modules from indecomposable Λ-modules when G is abelian. In particular, we apply this result to the deformed preprojective algebra Π Q λ , and get a reflection functor for the module category of Π Q λ G . Furthermore, we construct a new quiver Q G and prove that Π Q λ G is Morita equivalent to Π Q G η for some η.

On Regularity in Codimension One of Irreducible Components of Module Varieties

Let Λ be a tame quasi-tilted algebra and d the dimension vector of an indecomposable Λ-module. In the paper we prove that each irreducible component of the variety of Λ-modules of dimension vector d is regular in codimension one.

The existence of maximal n-orthogonal subcategories

For an ( n − 1 ) -Auslander algebra Λ with global dimension n, we give some necessary conditions for Λ admitting a maximal ( n − 1 ) -orthogonal subcategory in terms of the properties of simple Λ-modules with projective dimension n − 1 or n. For an almost hereditary algebra Λ with global dimension 2, we prove that Λ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable Λ-module M, M is injective is equivalent to that the reduced grade of M is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence of maximal n-orthogonal subcategories of T ⊥ for a cotilting module T. For a Gorenstein algebra, we prove that all non-projective direct summands of a maximal n-orthogonal module are Ω n τ -periodic. In addition, we study the relation between the complexity of modules and the existence of maximal n-orthogonal subcategories for the tensor product of two finite-dimensional algebras.

Preprojective Cluster Variables of Acyclic Cluster Algebras

It is proved that any cluster-tilted algebra defined in the cluster category 𝒞(H) has the same representation type as the initial hereditary algebra H. For any valued quiver (Γ, Ω), an injection from the subset 𝒫ℐ(Ω) of the cluster category 𝒞(Ω) consisting of indecomposable preprojective objects, preinjective objects, and the first shifts of indecomposable projective modules to the set of cluster variables of the corresponding cluster algebra 𝒜Ω is given. The images are called “preprojective cluster variables”. It is proved that all preprojective cluster variables other than ui have denominators u dim M in their irreducible fractions of integral polynomials, where M is the corresponding preprojective module or preinjective module. In case the valued quiver (Γ, Ω) is of finite type, the denominator theorem holds with respect to any cluster. Namely, let x = (x1,…, xn) be a cluster of the cluster algebra 𝒜Ω, and V the cluster tilting object in 𝒞(Ω) corresponding to x, whose endomorphism algebra is denoted by Λ. Then the denominator of any cluster variable y other than xi is x dim M, where M is the indecomposable Λ-module corresponding to y. This result is a generalization of the corresponding result of Caldero–Chapoton–Schiffler to the non-simply-laced case.

On Hom-spaces of tame algebras

Abstract Let Λ be a finite dimensional algebra over an algebraically closed field k and Λ has tame representation type. In this paper, the structure of Hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated Λ-modules and generic Λ-modules. In particular, such spaces are essentially controlled by those of the corresponding generic modules.

The Gabriel–Roiter measure for representation-finite hereditary algebras

Assume Λ is a representation-finite hereditary algebra over an algebraically closed field k. We provide a method to find orthogonal exceptional pairs for an indecomposable Λ-module and study the almost split sequences terminating at a Gabriel–Roiter factor module.

A Note on Periodicity in Representation Theory of Finite Dimensional Algebras

Let Λ be an algebra over an algebraically closed field K of infinite transcendence degree over its prime subfield. We prove in particular that if every s-dimensional indecomposable nonprojective Λ-module is Ω-periodic (resp. DTr-periodic) then there exists a common bound for the Ω-periods (resp. DTr-periods) of s-dimensional indecomposable nonprojective Λ-modules.

QUASITILTED EXTENSIONS BY NONDIRECTING MODULES

Let M be an indecomposable nondirecting module over a finite dimensional k-algebra Λ, and form the extension of Λ by the Λ–G-bimodule M, where G ⊆ EndΛ (M)op. We will show that if such an extension is quasitilted, then G = EndΛ (M)op and G is a division algebra. We also show that the double extension of Λ by M is not quasitilted, where G = EndΛ (M)op. So for every indecomposable nondirecting Λ-module M it is not possible to extend Λ with M more than once and still get a quasitilted algebra. If M is a nonzero decomposable Λ-module and is quasitilted, we then show that M is either directing or M has to be of a special form.

Non-split extensions over representation-finite hereditary algebras

Let be the path algebra of a quiver with underlying graph An or Dn. For all pairs of indecomposable Λ-modules M and N is computed and a basis consisting of short exact sequences 0 → M → Y → N → 0 is constructed.

Some Hall Polynomials for Representation-Finite Trivial Extension Algebras

Letkbe a finite field and assume that Λ is a finite dimensional associativek-algebra with 1. Denote by modΛ the category of all finitely generated (right) Λ-modules and by indΛ the full subcategory in which every object is a representative of the isoclass of an indecomposable (right) Λ-module. We are interested in the existance of the Hall polynomial ϕMNLfor anL,M,N∈modΛ (for the definition, see977113or Section 1 below). In case Λ is directed,7has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by4. It has been conjectured in8that any representation-finitek-algebra has Hall polynomials. In this investigation, we shall show that if Λ is a representation-finite trivial extension algebra, then, for anyL,M,N∈modΛ withNindecomposable, Λ has the Hall polynomials ϕMLNand ϕMNL. Using these Hall polynomials, we can naturally structure the free abelian group with a basis indΛ, denoted byK(modΛ), into a Lie algebra and the universal enveloping algebra ofK(modΛ)⊗ZQis just H(Λ)1⊗ZQ, where H(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below).

Almost Split Sequences whose Middle Term has at most Two Indecomposable Summands

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know thatthere exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].