On the classification of higher Auslander algebras for Nakayama algebras (original) (raw)

On bounds of homological dimensions in Nakayama algebras

Proceedings of the American Mathematical Society, Series B

Let A be a Nakayama algebra with n simple modules and a simple module S of even projective dimension. Choose m minimal such that a simple A-module with projective dimension 2m exists. Then we show that the global dimension of A is bounded by n + m − 1. This gives a combined generalisation of results of Gustafson [

Projective dimensions and Nakayama algebras

"Representations of algebras and related topics"

Institutt for matematiske fag, NTNU NO-7491 Trondheim Norway dagma@math.ntnu.no Dedicated to Professor Vlastimil Dlab on the occation of his sixtieth birthday.

A new characterisation of quasi-hereditary Nakayama algebras and applications

2021

We call a finite dimensional algebra A S-connected if the projective dimensions of the simple A-modules form an intervall. We prove that a Nakayama algebra A is S-connected if and only if A is quasi-hereditary. We apply this result to improve an inequality for the global dimension of quasihereditary Nakayama algebras due to Brown. We furthermore classify the Nakayama algebras where equality is attained in Brown’s inequality and show that they are enumerated by the even indexed Fibonacci numbers if algebra is cyclic and by the odd indexed Fibonacci numbers if algebra is linear.

The φ-dimension of cyclic Nakayama algebras

Communications in Algebra, 2021

K. Igusa and G. Todorov introduced the ϕ function which generalizes the notion of projective dimension. We study the behavior of the ϕ function for cyclic Nakayama algebras of infinite global dimension. We prove that the supremum of values of ϕ is always an even number. In particular we show that the ϕ-dimension is 2 if and only if the algebra satisfies certain symmetry conditions. Also we give a sharp upper bound for ϕ-dimension in terms of the number of monomial relations which describes the algebra.

Higher Auslander algebras admitting trivial maximal orthogonal subcategories

Journal of Algebra, 2011

For an Artinian (n − 1)-Auslander algebra Λ with global dimension n(≥ 2), we show that if Λ admits a trivial maximal (n − 1)-orthogonal subcategory of mod Λ, then Λ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in mod Λ is at most n − 1. As a result, for an Artinian Auslander algebra with global dimension 2, if Λ admits a trivial maximal 1-orthogonal subcategory of mod Λ, then Λ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra Λ over an algebraically closed field K, we show that Λ is a basic and connected (n − 1)-Auslander algebra Λ with global dimension n(≥ 2) admitting a trivial maximal (n − 1)-orthogonal subcategory of mod Λ if and only if Λ is given by the quiver:

Global Dimension of Some Artinian Algebras

arXiv preprint arXiv:1206.3726, 2012

The structure of arbitrary associative commutative unital artinian algebras is well-known: they are finite products of associative commutative unital local algebras pg.351, Cor. 23.12]. In the semi-simple case, we have the Artin-Wedderburn Theorem which states that any semi-simple artinian algebra (which is assumed to be associative and unital but not necessarily commutative) is a direct product of matrix algebras over division rings pg.35, Par. 3.5]. Along these lines, we observe a simple classification of artinian algebras and their representations in Proposition 1.3.2 (hereby referred as the Classification Lemma) in terms of a category in which each object has a local artinian endomorphism algebra. This category is constructed using a fixed set of primitive (not necessarily central) idempotents in the underlying algebra. The Classification Lemma is a version of Freyd's Representation Theorem [4, Sect. 5.3]: from an artinian algebra A we create a category C A on finitely many objects, and then the category of A-modules can be realized as a category of functors which admit C A as their domain. This construction can also be thought as a higher dimensional analogue of the semi-trivial extensions of [10] for artinian algebras.

Right n-Nakayama Algebras and their Representations

Algebras and Representation Theory, 2019

In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra Λ is representation-finite if and only if Λ is right n-Nakayama for some positive integer n. We classify hereditary right n-Nakayama algebras. We also define right n-coNakayama algebras and show that an artin algebra Λ is right n-coNakayama if and only if Λ is left n-Nakayama. We then study right 2-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right 2-Nakayama algebra. We end by classifying finite dimensional right 2-Nakayama algebras in terms of their quivers with relations. 1. Introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as an R-module. Given an artin algebra Λ, it is a quite natural question to ask for the classification of all the indecomposable finitely generated right Λ-modules. Only for few classes of algebras such a classification is known, one of the first such class were the Nakayama algebras. A Nakayama algebra Λ is an algebra such that the indecomposable projective right Λ-modules as well as the indecomposable injective right Λ-modules are uniserial. This then implies that all the indecomposable right Λmodules are uniserial. Nakayama algebras were studied by Tadasi Nakayama who called them generalized uniserial rings [9, 10]. A right Λ-module M is called uniserial if it has a unique composition series. Uniserial modules are the simplest indecomposables and this makes it interesting to understand their role in the category mod(Λ) of finitely generated right Λ-modules. An artin algebra Λ is said to be representation-finite, provided there are only finitely many isomorphism classes of indecomposable right Λ-modules. In representation theory, representation-finite algebras are of particular importance since in this case one has a complete combinatorial description of the module category in terms of the Auslander-Reiten quiver. The class of Nakayama algebras is one of the fundamental classes of representation-finite algebras whose representation theory is completely understood. In this paper we introduce the notion of n-factor serial modules. We say that a nonuniserial right Λ-module M of length l is n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. In some sense, n is an invariant that measures how far M is from being uniserial. We say that an artin algebra Λ is right n-Nakayama if every finitely generated indecomposable right Λ-module is i-factor serial for some 1 i n

Global Dimensions of Some Artinian Algebras

In this article we obtain lower and upper bounds for global dimensions of a class of artinian algebras in terms of global dimensions of a finite subset of their artinian subalgebras. Finding these bounds for the global dimension of an artinian algebra AAA is realized via an explicit algorithm we develop. This algorithm is based on a directed graph (not the Auslander-Reiten quiver) we construct, and it allows us to decide whether an artinian algebra has finite global dimension in good number of cases.

Representations of right 3-Nakayama algebras

arXiv (Cornell University), 2018

In this paper we study the category of finitely generated modules over a right 3-Nakayama artin algebra. First we give a characterization of right 3-Nakayama artin algebras and then we give a complete list of non-isomorphic finitely generated indecomposable modules over any right 3-Nakayama artin algebra. Also we compute all almost split sequences for the class of right 3-Nakayama artin algebras. Finally, we classify finite dimensional right 3-Nakayama algebras in terms of their quivers with relations. 1. introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as a R-module. Let Λ be an artin algebra. A right Λ-module M is called uniserial (1-factor serial) if it has a unique composition series. An artin algebra Λ is called Nakayama algebra if any indecomposable right Λ-module is uniserial. The class of Nakayama algebras is one the important class of representation finite algebras whose representation theory completely understood [3]. According to [5, Definition 2.1], a non-uniserial right Λ-module M of length l is called n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. An artin algebra Λ is called right n-Nakayama if every indecomposable right Λ-module is i-factor serial for some 1 ≤ i ≤ n and there exists at least one indecomposable n-factor serial right Λ-module [5, Definition 2.2]. The authors in [5] showed that the class of right n-Nakayama algebras provide a nice partition of the class of representation finite artin algebras. More precisely, the authors proved that an artin algebra Λ is representation finite if and only if Λ is right n-Nakayama for some positive integer n [5, Theorem 2.18]. The first part of this partition is the class of Nakayama algebras and the second part is the class of right 2-Nakayama algebras. Indecomposable modules and almost split sequences for the class of right 2-Nakayama algebras are classified in section 5 of [5]. In this paper we will study the class of right 3-Nakayama algebras. We first show that an artin algebra Λ which is neither Nakayama nor right 2-Nakayama is right 3-Nakayama if and only if every indecomposable right Λ-module of length greater than 4 is uniserial and every indecomposable right Λmodule of length 4 is local. Then we classify all indecomposable modules and almost split sequences over a right 3-Nakayama artin algebra. We also show that finite dimensional right 3-Nakayama algebras are special biserial and we describe all finite dimensional right 3-Nakayama algebras by their quivers and relations. Riedtmann in [6] and [7], by using the covering theory, classified representation-finite self-injective algebras.