Auslander Reiten Theory Research Articles

Simples in a cotilting heart

Every cotilting module over a ring R induces a t-structure with a Grothendieck heart in the derived category D(Mod-R). We determine the simple objects in this heart and their injective envelopes, combining torsion-theoretic aspects with methods from the model theory of modules and Auslander-Reiten theory.

Auslander–Reiten theory in extriangulated categories

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category C _ \underline {\mathscr {C}} of an extriangulated category C \mathscr {C} is a τ \tau -category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if C \mathscr {C} has enough projectives, almost split extensions and source morphisms. This gives various consequences on C _ \underline {\mathscr {C}} , including Igusa–Todorov’s Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of C _ \underline {\mathscr {C}} via the complete mesh category of the Auslander–Reiten species of C _ \underline {\mathscr {C}} . Finally we prove that any locally finite symmetrizable τ \tau -quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms.

Ideal mutations in triangulated categories and generalized Auslander-Reiten theory

We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino [15] by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom-finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen [17, Theorem 3.3] to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J: (i) J is functorially finite, (ii) GhJ=CoGhJ, and (iii) GhJ-source maps coincide with GhJ-sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations.

Determination of some almost split sequences in morphism categories

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams.

Representation Theory of Geigle-Lenzing Complete Intersections

Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings R R graded by abelian groups L \mathbb {L} of rank 1 1 , which we call Geigle-Lenzing complete intersections. We study the stable category C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R of Cohen-Macaulay representations, which coincides with the singularity category D s g L ( R ) \mathsf {D}^{\mathbb {L}}_{\mathrm {sg}}(R) . We show that C M _ L R \underline {\mathsf {CM}}^{\mathbb {L}}R is triangle equivalent to D b ( m o d A C M ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{\mathrm {CM}}) for a finite dimensional algebra A C M A^{\mathrm {CM}} , which we call the CM-canonical algebra. As an application, we classify the ( R , L ) (R,\mathbb {L}) that are Cohen-Macaulay finite. We also give sufficient conditions for ( R , L ) (R,\mathbb {L}) to be d d -Cohen-Macaulay finite in the sense of higher Auslander-Reiten theory. Secondly, we study a new class of non-commutative projective schemes in the sense of Artin-Zhang, i.e. the category c o h X = m o d L R / m o d 0 L R \mathsf {coh}\mathbb {X}=\mathsf {mod}^{\mathbb {L}}R/\mathsf {mod}^{\mathbb {L}}_0R of coherent sheaves on the Geigle-Lenzing projective space X \mathbb {X} . Geometrically this is the quotient stack X = [ X / G ] \mathbb {X}=[X/G] for X = S p e c R ∖ { R + } X={\mathrm {Spec}}\,R\setminus \{R_+\} and G = S p e c k [ L ] G={\mathrm {Spec}}\,k[\mathbb {L}] . We show that D b ( c o h X ) \mathsf {D}^{\mathrm {b}}(\mathsf {coh}\mathbb {X}) is triangle equivalent to D b ( m o d A c a ) \mathsf {D}^{\mathrm {b}}(\mathsf {mod} A^{ca}) for a finite dimensional algebra A c a A^{\mathrm {ca}} , which we call a d d -canonical algebra. We study when X \mathbb {X} is d d -vector bundle finite, and when X \mathbb {X} is derived equivale’nt to a d d -representation infinite algebra in the sense of higher Auslander-Reiten theory. Our d d -canonical algebras provide a rich source of d d -Fano and d d -anti-Fano algebras in non-commutative algebraic geometry. We also observe Orlov-type semiorthogonal decompositions of D s g L ( R ) \mathsf {D}_{\mathrm {sg}}^{\mathbb {L}}(R) and D b ( c o h X ) \mathsf {D}^{b}(\mathsf {coh}\mathbb {X}) .

Morphisms determined by objects under Galois G-covering theory

The theory of morphisms being determined by objects was originally investigated by Auslander, and can be seen as the culmination part of Auslander-Reiten theory. This theory provides a more general frame for working with the Auslander-Reiten theory. In this paper, we will study the behavior of morphisms determined by objects under G-coverings in the sense of Asashiba, which is a generalization of Gabriel's Galois coverings. As an application, we reformulate Bautista and Liu's framework that normal G-coverings preserve sink maps and source maps. We also show that there is a G-covering between two relative stable categories, which unifies Asashiba, Hafezi, Vahed and Mahdavi's work on the stable categories. This is applied to the discussion on the existence of Serre functors by G-coverings of triangulated categories.

Auslander's formula and correspondence for exact categories

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category modadm(E) of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category E. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of E are reflected in modadm(E), for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe modadm(E) as a subcategory of mod(E) when E is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use modadm(E) to give a bijection between exact structures on an idempotent complete additive category C and certain resolving subcategories of mod(C).

A bijection triangle in extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories, and recently Iyama, Nakaoka and Palu investigated Auslander-Reiten theory in terms of Auslander-Reiten-Serre duality in extriangulated categories. In this paper, we introduce the notion of Serre duality, as a special type of Auslander-Reiten-Serre duality, and then give an equivalent condition for the existence of Serre duality. On the other hand, we study a bijection triangle in extriangulated categories which involves the restricted Auslander bijection and Auslander-Reiten-Serre duality. Let R be a commutative artinian ring. We show that the restricted Auslander bijection holds true in Hom-finite R-linear Krull-Schmidt extriangulated categories having Auslander-Reiten-Serre duality, and especially obtain the Auslander bijection in Hom-finite R-linear Krull-Schmidt extriangulated categories having Serre duality. We also give some applications, and in particular, we show that a conjecture given by Ringel holds true in this setting.

Abelian quotients of categories of short exact sequences

We mainly investigate abelian quotients of categories of short exact sequences. The natural framework to consider the question is via identifying quotients of morphism categories as module categories. These ideas not only can be used to recover the abelian quotients produced by cluster-tilting subcategories of both exact categories and triangulated categories, but also can be used to reach our goal. Let (C,E) be an exact category. We denote by E(C) the category of bounded complexes whose objects are given by short exact sequences in E and by SE(C) the full subcategory formed by split short exact sequences. In general, E(C) is just an exact category, but the quotient E(C)/[SE(C)] turns out to be abelian. In particular, if (C,E) is Frobenius, we present three equivalent abelian quotients of E(C) and point out that the equivalences are actually given by left and right rotations. The abelian quotient E(C)/[SE(C)] admits some nice properties. We explicitly describe the abelian structure, projective objects, injective objects and simple objects, which provide a new viewpoint to understanding Hilton-Rees Theorem and Auslander-Reiten theory. Furthermore, we present some analogous results for triangulated versions.

D-Representation-finite self-injective algebras

In this paper, we initiate the study of higher-dimensional Auslander–Reiten theory of self-injective algebras. We give a systematic construction of (weakly) d-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most d. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for k-linear categories to the setting of higher-dimensional Auslander–Reiten theory.Applications include n-fold trivial extensions and (classical and higher) preprojective algebras, which are shown to be d-representation-finite in many cases. We also get a complete classification of all d-representation-finite self-injective Nakayama algebras for arbitrary d.

The Nakayama automorphism of a self‐injective preprojective algebra

We give a simple proof, using Auslander-Reiten theory, that the preprojective algebra of a basic hereditary algebra of finite representation type is Frobenius. We then describe its Nakayama automorphism, which is induced by the Nakayama functor on the module category of our hereditary algebra.

Simplicial structures in higher Auslander–Reiten theory

We develop a novel combinatorial perspective on the higher Auslander algebras of type A, a family of algebras arising in the context of Iyama's higher Auslander–Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg–MacLane spaces and higher-dimensional versions of Waldhausen's S•-construction in algebraic K-theory. As an application of our techniques we provide a generalisation of the higher reflection functors of Iyama and Oppermann to representations with values in stable ∞-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type A quivers developed by Groth and Šťovíček. Our simplicial point of view then naturally leads to an interplay between slice mutation, horn filling conditions, and the higher Segal conditions of Dyckerhoff and Kapranov. In this context, we provide a classification of higher Segal objects with values in any abelian category or stable ∞-category.

Higher Nakayama algebras I: Construction

We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander–Reiten theory. More precisely, for each Nakayama algebra A and each positive integer d, we construct a finite dimensional algebra A(d) having a distinguished d-cluster-tilting A(d)-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of A. We also construct higher dimensional analogues of the mesh category of type ZA∞ and the tubes.

Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories

We generalise some of the theory developed for abelian categories in papers of Auslander and Reiten to semi-abelian and quasi-abelian categories. In addition, we generalise some Auslander-Reiten theory results of S. Liu for Krull-Schmidt categories by removing the Hom-finite and indecomposability restrictions. Finally, we give equivalent characterisations of Auslander-Reiten sequences in a quasi-abelian, Krull-Schmidt category.

Specifying the Auslander transpose in submodule category and its applications

Let (R,m) be a d-dimensional commutative Noetherian local ring. Let M denote the morphism category of finitely generated R-modules, and let S be the full subcategory of M consisting of monomorphisms, known as the submodule category. This article reveals that the Auslander transpose in the category S can be described explicitly within modR, the category of finitely generated R-modules. This result is exploited to study the linkage theory as well as the Auslander–Reiten theory in S. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander–Reiten translations in the subcategories H and G, consisting of all morphisms which are maximal Cohen–Macaulay R-modules and Gorenstein projective morphisms, respectively, may be computed within modR via G-covers. The corresponding result for the subcategory of epimorphisms in H is also obtained.

N-T-COTORSION-FREE MODULES

AbstractIn order to better unify the tilting theory and the Auslander–Reiten theory, Xi introduced a general transpose called the relative transpose. Originating from this, we introduce and study the cotranspose of modules with respect to a left A-module T called n-T-cotorsion-free modules. Also, we give many properties and characteristics of n-T-cotorsion-free modules under the help of semi-Wakamatsu-tilting modules AT.

An introduction to higher Auslander-Reiten theory

This article consists of an introduction to Iyama's higher Auslander-Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander-Reiten theory, including the existence of $d$-almost-split sequences in $d$-cluster-tilting subcategories, following the approach to classical Auslander-Reiten theory due to Auslander, Reiten, and Smal{\o}. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for $d$-exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in $d$-cluster-tilting subcategories.

Auslander–Gorenstein algebras and precluster tilting

We generalize the notions of n-cluster tilting subcategories and τ-selfinjective algebras into n-precluster tilting subcategories and τn-selfinjective algebras, where we show that a subcategory naturally associated to n-precluster tilting subcategories has a higher Auslander–Reiten theory. Furthermore, we give a bijection between n-precluster tilting subcategories and n-minimal Auslander–Gorenstein algebras, which is a higher dimensional analog of Auslander–Solberg correspondence [8] as well as a Gorenstein analog of n-Auslander correspondence [22]. The Auslander–Reiten theory associated to an n-precluster tilting subcategory is used to classify the n-minimal Auslander–Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander–Solberg.

Restriction and induction of indecomposable modules over the Temperley–Lieb algebras

Both the original Temperley–Lieb algebras and their dilute counterparts form families of filtered algebras: and , for all . For each such inclusion, the restriction and induction of every finite-dimensional indecomposable module over (or ) is computed. To accomplish this, a thorough description of each indecomposable is given, including its projective cover and injective hull, some short exact sequences in which it appears, its socle and head, and its extension groups with irreducible modules. These data are also used to prove the completeness of the list of indecomposable modules, up to isomorphism. In fact, two completeness proofs are given—the first is based on elementary homological methods and the second uses Auslander–Reiten theory. The latter proof offers a detailed example of this algebraic tool that may be of independent interest.

A criterion for graded coherence of tensor algebras and applications to higher dimensional AR-theory

Even if a ring A is coherent, the polynomial ring A[X] in one variable could fail to be coherent. In this note we show that A[X] is graded coherent with the grading \(\deg X=1\). More generally, we give a criterion of graded coherence of the tensor algebra Open image in new window of a certain class of bi-module \(\sigma \). As an application of the criterion, we show that there is a relationship between higher dimensional Auslander–Reiten theory and graded coherence of higher preprojective algebras. In the appendix by Srikanth B. Iyengar, we deal with a commutative Noetherian ring R and show that the class of R-R-bi-module to which the criterion is applicable is precisely the class of projective modules.