Auslander Reiten Translation Research Articles

Sincere silting modules and vanishing conditions

We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. Let R be a perfect ring and T be an R-module. It is proved that T is sincere silting if and only if T is presilting satisfing the vanishing condition KerExt R 0 ≤ i ≤ 1 ( T , − ) = 0 , and that T is tilting if and only if KerExt R 0 ⩽ i ⩽ 1 ( T , − ) = 0 and Gen T ⊆ KerExt R 1 ⩽ i ⩽ 2 ( T , − ) . As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if Ext R i ( T , T ( J ) ) = 0 for all sets J and all integer i ≥ 1 . This not only extends a main result of Zhang’s paper [Self-orthogonal τ-tilting modules and tilting modules, J Pure Appl Algebra, 2022, 226: 106860] from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using Auslander-Reiten translations.

When does the Auslander–Reiten translation operate linearly on the Grothendieck group? – Part I

For a hereditary, finite-dimensional algebra A the Coxeter transformation extends the action of the Auslander–Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the category of finitely generated A-modules. It is natural to ask whether other algebras admit a similar linear extension. We show that this is indeed the case for all Nakayama algebras. Conversely, we show that finite-dimensional algebras with non-acyclic and connected quiver admitting such a linear extension are already cyclic Nakayama algebras.

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.

On higher-dimensional Auslander–Reiten theory for comodule categories

We introduce the concepts of higher-dimensional defect functors and $n$-Auslander–Reiten translations. Then we prove the higher-dimensional Auslander’s defect formula for the category of finitely $n$-copresented comodules over a left semiperfect coalgebra

Serre Functors and Graded Categories

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.

Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let [Formula: see text] be a fixed algebraically closed field of arbitrary characteristic, let [Formula: see text] be a finite dimensional self-injective [Formula: see text]-algebra, and let [Formula: see text] be an indecomposable non-projective left [Formula: see text]-module with finite dimension over [Formula: see text]. We prove that if [Formula: see text] is the Auslander–Reiten translation of [Formula: see text], then the versal deformation rings [Formula: see text] and [Formula: see text] (in the sense of F.M. Bleher and the second author) are isomorphic. We use this to prove that if [Formula: see text] is further a cluster-tilted [Formula: see text]-algebra, then [Formula: see text] is universal and isomorphic to [Formula: see text].

Dynamical combinatorics and torsion classes

For finite semidistributive lattices the map κ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements. Here we study the κ -map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the κ -map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend κ to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll in 2019. For hereditary algebras , we show that the extended κ -map on torsion classes is essentially the same as Ringel's ϵ -map on wide subcategories . Also in the hereditary case, we relate the square of κ to the Auslander-Reiten translation.

CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS

The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.

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.

Paths to Understanding Birational Rowmotion on Products of Two Chains

Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset $P$, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to $Y$-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D.~Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.

The generalized Auslander–Reiten duality on an exact category

We introduce a notion of generalized Auslander–Reiten duality on a Hom-finite Krull–Schmidt exact category [Formula: see text]. This duality induces the generalized Auslander–Reiten translation functors [Formula: see text] and [Formula: see text]. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text]. A non-projective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the first term of an almost split conflation. We study the generalized Auslander–Reiten duality on the category of finitely presented representations of locally finite interval-finite quivers.

The Coxeter Transformation on Cominuscule Posets

Let $\mathsf{J(C)}$ be the poset of order ideals of a cominuscule poset $\mathsf{C}$ where $\mathsf{C}$ comes from two of the three infinite families of cominuscule posets or the exceptional cases. We show that the Auslander-Reiten translation $\tau$ on the Grothendieck group of the bounded derived category for the incidence algebra of the poset $\mathsf{J(C)}$, which is called the \emph{Coxeter transformation} in this context, has finite order. Specifically, we show that $\tau^{h+1}=\pm id$ where $h$ is the Coxeter number for the relevant root system.

Triangulated quotient categories revisited

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this paper. Let A be an extension closed subcategory of an extriangulated category C. Then the additive quotient category M:=A/[X] carries naturally a triangulated structure whenever (A,A) forms an X-mutation pair. This result generalizes many results of the same type for triangulated categories. It is used to give a classification of thick triangulated subcategories of pre-triangulated category C/[X], where X is functorially finite in C. When C has Auslander–Reiten translation τ, we prove that for a functorially finite subcategory X of C containing projectives and injectives, the quotient C/[X] is a triangulated category if and only if (C,C) is X-mutation, and if and only if τX_=X‾. This generalizes a result by Jørgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory X of the extriangulated category C, C admits a new extriangulated structure such that C is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present extriangulated categories which are neither exact categories nor triangulated categories.

Dual $F$-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities

The notion of $F$-signature is defined by C. Huneke and G. Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules and called dual $F$-signature. In this paper, we determine the dual $F$-signature of a certain class of Cohen-Macaulay modules (so-called "special") over cyclic quotient surface singularities. Also, we compare the dual $F$-signature of a special Cohen-Macaulay module with that of its Auslander-Reiten translation. This gives a new characterization of the Gorensteiness.

The closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one

Let Λ be a concealed canonical algebra and d the dimension vector of a Λ-module which is periodic respect to the action of the Auslander–Reiten translation τ. In the paper, we investigate the union of the closures of the orbits of the τ-periodic Λ-modules of dimension vector d. We show that this set is closed and regular in codimension one.

Cluster categories for marked surfaces: punctured case

We study cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^{1}$ as intersection numbers of tagged curves and Auslander–Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

Weighted noncommutative regular projective curves

Let \mathcal H be a noncommutative regular projective curve over a perfect field k . We study global and local properties of the Auslander–Reiten translation \tau and give an explicit description of the complete local rings, with the involvement of \tau . We introduce the \tau -multiplicity e_{\tau}(x) , the order of \tau as a functor restricted to the tube concentrated in x . We obtain a local-global principle for the (global) skewness s(\mathcal H) , defined as the square root of the dimension of the function (skew-) field over its centre. In the case of genus zero we show how the ghost group, that is, the group of automorphisms of \mathcal H which fix all objects, is determined by the points x with e_{\tau}(x) > 1 . Based on work of Witt we describe the noncommutative regular (smooth) projective curves over the real numbers; those with s(\mathcal H)=2 we call Witt curves. In particular, we study noncommutative elliptic curves, and present an elliptic Witt curve which is a noncommutative Fourier–Mukai partner of the Klein bottle. If \mathcal H is weighted, our main result will be formulae for the orbifold Euler characteristic, involving the weights and the \tau -multiplicities. As an application we will classify the noncommutative 2-orbifolds of nonnegative Euler characteristic, that is, the real elliptic, domestic and tubular curves. Throughout, many explicit examples are discussed.

Auslander–Reiten Sequences, Brown–Comenetz Duality, and the K(n)-local Generating Hypothesis

In this paper, we construct a version of Auslander–Reiten sequences for the K(n)-local stable homotopy category. In particular, the role of the Auslander–Reiten translation is played by the local Brown–Comenetz duality functor. As an application, we produce counterexamples to the K(n)-local generating hypothesis for all heights n > 0 and all primes. Furthermore, our methods apply to other triangulated categories, as for example the derived category of quasi-coherent sheaves on a smooth projective scheme.

On cluster-tilting objects in a triangulated category with Serre duality

ABSTRACTLet 𝒟 be a Krull–Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of an FΛ-stable support τ-tilting module, induced by the shift functor and the Auslander–Reiten translation, in the cluster-tilted algebra . We show that there exists a bijection between basic cluster-tilting objects in 𝒟 and basic FΛ-stable support τ-tilting Λ-modules. This generalizes a result of Adachi–Iyama–Reiten [1]. As a consequence, we obtain that all cluster-tilting objects in 𝒟 have the same number of nonisomorphic indecomposable direct summands.

On syzygies over 2-Calabi–Yau tilted algebras

We characterize the syzygies and co-syzygies over 2-Calabi–Yau tilted algebras in terms of the Auslander–Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen–Macaulay modules, the representation dimension of algebras and the Igusa–Todorov functions. In particular, we prove that the Igusa–Todorov dimensions of d-Gorenstein algebras are equal to d.For cluster-tilted algebras of Dynkin type D, we give a geometric description of the stable Cohen–Macaulay category in terms of tagged arcs in the punctured disc. We also describe the action of the syzygy functor in a geometric way. This description allows us to compute the Auslander–Reiten quiver of the stable Cohen–Macaulay category using tagged arcs and geometric moves.