Finite Representation Type Research Articles

On FGDF-modules

Let R be a unital ring and M a unitary module not necessary over R. The FGDF-module is a generalization of FGDF-rings (Touré, Diop, Mohamed and Sangharé, 2014). In this work, we first give some properties of FGDF-modules. After that, we show that for a finitely generated module M, M is a FGDF-module if and only if M is of finite representation type module. Finally, we show that M is a finitely generated FGDF-module if and only if every Dedekind finite module of $\sigma[M]$ is noetherian.

A function on the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to liaison theory

Let $(A,\mathfrak{m})$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\operatorname{\underline{CM}}(A)$ be the stable category of maximal Cohen-Macauley $A$-modules and let $\operatorname{\underline{ICM}}(A)$ denote the set of isomorphism classes in $\operatorname{\underline{CM}}(A)$. We define a function $\xi \colon \operatorname{\underline{ICM}}(A) \to \mathbb{Z}$ which behaves well with respect to exact triangles in $\operatorname{\underline{CM}}(A)$. We then apply this to (Gorenstein) liaison theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liaison classes of $\{\,\mathfrak{m}^n \mid n\geq 1 \,\}$ is an infinite set. We also prove that if $A$ is Henselian with finite representation type with $A/\mathfrak{m}$ uncountable then for each $m \geq 1$ the set $\mathcal {C}_m = \{\, I \mid I \text { is a codim $2$ CM-ideal with } e_0(A/I) \leq m \,\}$ is contained in finitely many even liaison classes $L_1,\dots ,L_r$ (here $r$ may depend on $m$).

Infinite dimensional representations of finite dimensional algebras and amenability

We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra R over a countable field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930’s. As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.

FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation

Some Results on FGS-modules

Let R be a commutative ring, with a unity 1 not equal to 0 and M a unitary left R-module. In this paper we give some properties of an FGS-module. After that we give others important characterizations. Indeed, we first show that M is a local FGS-module if and only if it is of finite representation type. Secondly, we show that M is a prime FGS-module if and only if it is a serial type module and of finite length if and only if it is a finite representation type module.

Higher Auslander Correspondence for Dualizing R-Varieties

Let R be a commutative artinian ring. We extend higher Auslander correspondence from Artin R-algebras of finite representation type to dualizing R-varieties. More precisely, for a positive integer d, we show that a dualizing R-variety is d-abelian if and only if it is a d-Auslander dualizing R-variety if and only if it is equivalent to a d-cluster-tilting subcategory of the category of finitely presented modules over a dualizing R-variety.

Desingularization of Quiver Grassmannians via Nakajima Categories

In this paper, we show that generalized Nakajima Categories provide a framework to construct a desingularization of quiver Grassmannians for self-injective algebras of finite representation type. Furthermore, we show that all standard Frobenius models of orbit categories of the bounded derived category considered in Keller, Documenta Math. 10: 551–581, 2005 are equivalent to proj 𝓡, the finitely generated projective modules of the regular Nakajima category 𝓡.

Relations for Grothendieck groups of Gorenstein rings

We consider the converse of the Butler, Auslander-Reiten's Theorem which is on the relations for Grothendieck groups. We show that a Gorenstein ring is of finite representation type if the Auslander-Reiten sequences generate the relations for Grothendieck groups. This gives an affirmative answer of the conjecture due to Auslander.

Cycles of modules and finite representation type

Cycles of modules and finite representation type

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

We prove that every cycle-finite artin algebra with finitely many isomorphism classes of τ-rigid indecomposable modules is of finite representation type.

Some computations of the generalized Hilbert-Kunz function and multiplicity

Let R R be a local ring of characteristic p > 0 p>0 which is F F -finite and has perfect residue field. We compute the generalized Hilbert-Kunz invariant for certain modules over several classes of rings: hypersurfaces of finite representation type, toric rings, and weakly F F -regular rings.

Gorenstein global dimension and Hopf algebroid actions

Let [Formula: see text] be a Hopf algebroid and [Formula: see text] a left [Formula: see text]-module algebra. In this paper, we mainly present the duality theorem for the smash product [Formula: see text], and making use of integral theory for Hopf algebroids, we investigate the stability of Gorenstein injective pre-envelopes and Gorenstein projective precovers between the category of [Formula: see text]-modules and the category of [Formula: see text]-modules. Moreover, we establish the relationship between Gorenstein global dimension of [Formula: see text] and that of [Formula: see text], and prove that [Formula: see text] has finite representation type, resp. is selfinjective, resp. is CM-finite [Formula: see text]-Gorenstein, if and only if [Formula: see text] has the same property under suitable conditions. As an application, we investigate the representation dimension of the lower triangular matrix Artin algebra [Formula: see text].

Homogeneous ACM bundles on a Grassmannian

In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM, for short) bundles on Grassmannians Gr(k,n) and we construct irreducible families of indecomposable ACM bundles on a Grassmannian Gr(k,n) of arbitrary high rank and dimension.As a consequence we prove that all Grassmann varieties Gr(k,n) are of wild representation type unless the projective spaces Gr(0,n) and Gr(n−1,n) and the hyperquadric Gr(1,3) in P5 which are of finite representation type.

Algebras of finite representation type arising from maximal rigid objects

We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi–Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are 2-Calabi–Yau-tilted, and therefore appear in an earlier classification by Bertani-Økland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi–Yau categories of finite type.

Degenerations of graded Cohen-Macaulay modules

We introduce a notion of degenerations of graded modules. In relation to it, we also introduce several partial orders as graded analogies of the hom order, the degeneration order and the extension order. We prove that these orders are identical on the graded Cohen-Macaulay modules over a graded ring which is of graded finite representation type and representation directed.

Noncommutative graded algebras of finite Cohen-Macaulay representation type

Let A be an AS-Cohen-Macaulay algebra. We show that if A is of finite CohenMacaulay representation type, then A is a noncommutative graded isolated singularity. This is a noncommutative analogue of a well-known theorem of Auslander and is a generalization of Jorgensen’s theorem. Besides, we give an example of a noncommutative quadric hypersurface of finite Cohen-Macaulay representation type in a quantum P which is not a domain. We also give all indecomposable graded maximal Cohen-Macaulay modules over it explicitly.

Degrees in Auslander–Reiten components with almost split sequences of at most two middle termss

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander–Reiten components Γ having only almost split sequences with at most two indecomposable middle terms, that is, α(Γ) ≤ 2. We prove that if f : X → Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z ∈ Γ and a morphism φ ∈ ℜn(Z, X)\ℜn+1(Z, X) for some positive integer n such that fφ = 0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f = (f1, f2)t : X → Y1 ⊕ Y2 is an irreducible epimorphism of finite left degree then we show that dl(f) = dl(f1) + dl(f2). We also characterize the artin algebras of finite representation type with α(ΓA) ≤ 2 in terms of a finite number of irreducible morphisms with finite degree.

Representations of modular skew group algebras

In this paper we study representations of skew group algebras $\Lambda G$, where $\Lambda$ is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field $k$ with characteristic $p \geqslant 0$, and $G$ is an arbitrary finite group each element of which acts as an algebra automorphism on $\Lambda$. We characterize skew group algebras with finite global dimension or finite representation type, and classify the representation types of transporter categories for $p \neq 2,3$. When $\Lambda$ is a locally finite graded algebra and the action of $G$ on $\Lambda$ preserves grading, we show that $\Lambda G$ is a generalized Koszul algebra if and only if so is $\Lambda$.

The geometry of finite dimensional algebras with vanishing radical square

Let Λ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical J vanishes. We determine the irreducible components of the module variety Repd(Λ) for any dimension vector d. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when Λ is local extends existing counts for the two-loop quiver to quivers with arbitrary finite sets of loops.For any algebra Λ with J2=0, our criteria for identifying the components of Repd(Λ) permit us to characterize the modules parametrized by the individual irreducible components. Focusing on such a component, we explore generic properties of the corresponding modules by establishing a geometric bridge between the algebras with zero radical square on the one hand and their stably equivalent hereditary counterparts on the other. The bridge links certain closed subvarieties of Grassmannians parametrizing the modules with fixed top over the two types of algebras. By way of this connection, we transfer results of Kac and Schofield from the hereditary case to algebras of Loewy length 2. Finally, we use the transit of information to show that any algebra of Loewy length 2 which enjoys the dense orbit property in the sense of Chindris, Kinser and Weyman has finite representation type.

On selfinjective algebras of finite representation type with maximal almost split sequences

We investigate the structure of finite dimensional selfinjective algebras of finite representation type over an arbitrary field having almost split sequences of modules whose middle term admits the maximal possible number of indecomposable direct summands.