Auslander Reiten Sequences Research Articles

Higher Auslander-Reiten sequences via morphisms determined by objects

Let be an Ext-finite, Krull-Schmidt and k-linear n-exangulated category with k a commutative artinian ring. In this note, we define two additive subcategories and of in terms of the representable functors from the stable category of to the category of finitely generated k-modules. Moreover, we show that there exists an equivalence between the stable categories of these two full subcategories. Finally, we give some equivalent characterizations on the existence of Auslander-Reiten n-exangles via determined morphisms. These results unify and extend their works by Jiao–Le for exact categories, Zhao–Tan–Huang for extriangulated categories, Xie–Liu–Yang for n-abelian categories.

Hammocks to visualize the support of finitely presented functors

Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered as a module over a subalgebra. When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor. The key tool is the Cokernel Complex Lemma which links the values of the hammock function to the Auslander-Reiten structure of the category. We are also interested in exact subcategories of module categories which have Auslander-Reiten sequences. Our examples include quiver representations and invariant subspaces of nilpotent linear operators.

On Irreducible Morphisms and Auslander-Reiten Triangles in the Stable Category of Modules over Repetitive Algebras

Let ${\Bbbk }$ be an algebraically closed field, let Λ be a finite dimensional ${\Bbbk }$ -algebra, and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. For the stable category of finitely generated left $\widehat {\Lambda }$ -modules $\widehat {\Lambda }$ -mod, we prove that every Auslander-Reiten triangle in $\widehat {\Lambda }$ -mod is induced from an Auslander-Reiten sequence of finitely generated left $\widehat {\Lambda }$ -modules. We use this fact to show that the irreducible morphisms in $\widehat {\Lambda }$ -mod fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. Finally, we use this classification to describe the shape of the Auslander-Reiten triangles in $\widehat {\Lambda }$ -mod.

Triangular matrix categories II: Recollements and functorially finite subcategories

In this paper we continue the study of triangular matrix categories $\mathbf {{\varLambda }}=\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]$ initiated in León-Galeana et al. (2022). First, given a additive category $\mathcal {C}$ and an ideal $\mathcal {I}_{{\mathscr{B}}}$ in $\mathcal {C}$ , we prove a well known result that there is a canonical recollement We show that given a recollement between functor categories we can induce a new recollement between triangular matrix categories, this is a generalization of a result given by Chen and Zheng in (J. Algebra, 321 (9), 2474–2485 2009, [Theorem 4.4]). In the case of dualizing K-varieties we can restrict the recollement we obtained to the categories of finitely presented functors. Given a dualizing variety $\mathcal {C}$ , we describe the maps category of $\text {mod}(\mathcal {C})$ as modules over a triangular matrix category and we study its Auslander-Reiten sequences and contravariantly finite subcategories, in particular we generalize several results from Martínez-Villa and Ortíz-Morales (Inter. J Algebra, 5 (11), 529–561 2011). Finally, we prove a generalization of a result due to Smalø (2011, [Theorem 2.1]), which give us a way of construct functorially finite subcategories in the category $\text {Mod}\Big (\left [\begin {smallmatrix} \mathcal {T} & 0 \\ M & \mathcal {U} \end {smallmatrix}\right ]\Big )$ from those of $\text {Mod}(\mathcal {T})$ and $\text {Mod}(\mathcal {U})$ .

The existence of n-Auslander–Reiten sequences via determined morphisms

Let [Formula: see text] be a commutative artinian ring and [Formula: see text] a small Ext-finite Krull–Schmidt [Formula: see text]-abelian [Formula: see text]-category with enough projectives and injectives. We introduce two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text] in terms of the representable functors from the stable category of [Formula: see text] to category of finitely generated [Formula: see text]-modules. Moreover, we define two additive functors [Formula: see text] and [Formula: see text], which are mutually quasi-inverse equivalences between the stable categories of this two full subcategories. We give an equivalent characterization on the existence of [Formula: see text]-Auslander–Reiten sequences using determined morphisms.

D-Auslander–Reiten sequences in subcategories

AbstractLet Φ be a finite-dimensional algebra over a field k. Kleiner described the Auslander–Reiten sequences in a precovering extension closed subcategory ${\rm {\cal X}}\subseteq {\rm mod }\,\Phi $. If $X\in \mathcal {X}$ is an indecomposable such that ${\rm Ext}_\Phi ^1 (X,{\rm {\cal X}})\ne 0$ and $\zeta X$ is the unique indecomposable direct summand of the $\mathcal {X}$-cover $g:Y\to D\,{\rm Tr}\,X$ such that ${\rm Ext}_\Phi ^1 (X,\zeta X)\ne 0$, then there is an Auslander–Reiten sequence in $\mathcal {X}$ of the form $${\rm \epsilon }:0\to \zeta X\to {X}^{\prime}\to X\to 0.$$Moreover, when ${\rm En}{\rm d}_\Phi (X)$ modulo the morphisms factoring through a projective is a division ring, Kleiner proved that each non-split short exact sequence of the form $$\delta :0\to Y\to {Y}^{\prime}\buildrel \eta \over \longrightarrow X\to 0$$is such that η is right almost split in $\mathcal {X}$, and the pushout of δ along g gives an Auslander–Reiten sequence in ${\rm mod}\,\Phi $ ending at X.In this paper, we give higher-dimensional generalizations of this. Let $d\geq 1$ be an integer. A d-cluster tilting subcategory ${\rm {\cal F}}\subseteq {\rm mod}\,\Phi $ plays the role of a higher ${\rm mod}\,\Phi $. Such an $\mathcal {F}$ is a d-abelian category, where kernels and cokernels are replaced by complexes of d objects and short exact sequences by complexes of d + 2 objects. We give higher versions of the above results for an additive ‘d-extension closed’ subcategory $\mathcal {X}$ of $\mathcal {F}$.

A Frobenius category within the monomorphism category

Given a selfinjective artin algebra Λ, we consider the category Sinj(Λ) of all embeddings of a left Λ-module in a finitely generated injective left Λ-module. We show that Sinj(Λ) is a Frobenius category with Auslander-Reiten sequences such that the categories Λ-mod and Sinj(Λ) are stably equivalent and Sinj(Λ) has twice as many indecomposable injective objects as Λ-mod.

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.

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.

Separated monic representations II: Frobenius subcategories and RSS equivalences

This paper aims at looking for Frobenius subcategories, via the separated monomorphism category ${\rm smon}(Q, I, \x)$, and on the other hand, to establish an {\rm RSS} equivalence from ${\rm smon}(Q, I, \x)$ to its dual ${\rm sepi}(Q, I, \x)$. For a bound quiver $(Q, I)$ and an algebra $A$, where $Q$ is acyclic and $I$ is generated by monomial relations, let $\Lambda=A\otimes_k kQ/I$. For any additive subcategory $\x$ of $A$-mod, we construct ${\rm smon}(Q, I, \x)$ combinatorially. This construction describe Gorenstein-projective $\m$-modules as $\mathcal {GP}(\m) = {\rm smon}(Q, I, \mathcal {GP}(A))$. It admits a homological interpretation, and enjoys a reciprocity ${\rm smon}(Q, I, ^\bot T)= ^\bot (T\otimes kQ/I)$ for a cotilting $A$-module $T$. As an application, ${\rm smon}(Q, I, \x)$ has Auslander-Reiten sequences if $\x$ is resolving and contravariantly finite with $\widehat{\x}=A$-mod. In particular, ${\rm smon}(Q, I, A)$ has Auslander-Reiten sequences. It also admits a filtration interpretation as ${\rm smon}(Q, I, \mathscr{X})={\rm Fil}(\mathscr{X}\otimes \mathcal P(kQ/I))$, provided that $\x$ is extension-closed. As an application, ${\rm smon}(Q, I, \x)$ is an extension-closed Frobenius subcategory if and only if so is $\x$. This gives new Frobenius subcategories of $\m$-mod in the sense that they are not $\mathcal{GP}(\m)$. Ringel-Schmidmeier-Simson equivalence ${\rm smon}(Q, I, \x)\cong{\rm sepi}(Q, I, \x)$ is introduced and the existence is proved for arbitrary extension-closed subcategories $\x$. In particular, the Nakayama functor $\mathcal N_\m$ gives an {\rm RSS} equivalence ${\rm smon}(Q, I, A)\cong{\rm sepi}(Q, I, A)$ if and only if $A$ is Frobenius. For a chain $Q$ with arbitrary $I$, an explicit formula of an {\rm RSS} equivalence is found for arbitrary additive subcategories $\x$.

Approximations, ghosts and derived equivalences

AbstractApproximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weaklyn-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.

Bimodule monomorphism categories and RSS equivalences via cotilting modules

The monomorphism category M(A,M,B) induced by a bimodule MBA is the subcategory of Λ-mod consisting of [XY]ϕ such that ϕ:M⊗BY→X is a monic A-map, where Λ=[AM0B], and A, B are Artin algebras. In general, M(A,M,B) is not the monomorphism category induced by quivers. It could describe the Gorenstein-projective Λ-modules. This monomorphism category is a resolving subcategory of Λ-mod if and only if MB is projective. In this case, it has enough injective objects and Auslander–Reiten sequences, and can be also described as the left perpendicular category of a unique basic cotilting Λ-module. If M satisfies the condition (IP) (see Subsection 1.6), then the stable category of M(A,M,B) admits a recollement of additive categories, which is in fact a recollement of singularity categories if M(A,M,B) is a Frobenius category. Ringel–Schmidmeier–Simson equivalence between M(A,M,B) and its dual is introduced. If M is an exchangeable bimodule, then an RSS equivalence is given by a Λ–Λ bimodule which is a two-sided cotilting Λ-module with a special property; and the Nakayama functor NΛ gives an RSS equivalence if and only if both A and B are Frobenius algebras.

On Auslander–Reiten sequences for Borel–Schur algebras

We classify Borel–Schur algebras having finite representation type. We also determine Auslander–Reiten sequences for a large class of simple modules over Borel–Schur algebras. A partial information on the structure of the socles of Borel-Schur algebras is given.

Syzygies of Cohen–Macaulay modules and Grothendieck groups

We study the converse of a theorem of Butler and Auslander–Reiten. We show that a Cohen–Macaulay local ring with an isolated singularity has only finitely many isomorphism classes of indecomposable summands of syzygies of Cohen–Macaulay modules if the Auslander–Reiten sequences generate the relation of the Grothendieck group of finitely generated modules. This extends a recent result of Hiramatsu, which gives an affirmative answer in the Gorenstein case to a conjecture of Auslander.

Recollements of derived categories III: finitistic dimensions

In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel’s reduction techniques for finitistic dimenson conjecture to more general contexts, but also generalise some recent results in the literature.

A note on Gorenstein transposes

Let [Formula: see text] be a left and right Noetherian ring. In this paper, we prove that any Gorenstein transpose of a finitely generated [Formula: see text]-module is exactly an Auslander transpose. As applications, we obtain a new relation between a Gorenstein transpose of a module with a transpose of the same module, and show that the Gorenstein transpose of a module is unique up to Gorenstein projective equivalence. In addition, when [Formula: see text] is an Artin algebra, the corresponding Auslander–Reiten sequences are constructed in terms of Gorenstein transposes.

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.

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.

Higher Auslander–Reiten sequences and t-structures

Let R be an artin algebra and C an additive subcategory of mod(R). We construct a t-structure on the homotopy category K−(C) and argue that its heart HC is a natural domain for higher Auslander–Reiten (AR) theory. In the paper [5] we showed that K−(mod(R)) is a natural domain for classical AR theory. Here we show that the abelian categories Hmod(R) and HC interact via various functors. If C is functorially finite then HC is a quotient category of Hmod(R). We illustrate our theory with two examples:When C is a maximal n-orthogonal subcategory Iyama developed a higher AR theory, see [10]. In this case we show that the simple objects of HC correspond to Iyama's higher AR sequences and derive his higher AR duality from the existence of a Serre functor on the derived category Db(HC).The category O of a complex semi-simple Lie algebra fits into higher AR theory in the situation when R is the coinvariant algebra of the Weyl group.

Auslander-Reiten Sequences or Triangles Related to Rigid Subcategories

Let 𝒞 be a triangulated category which has Auslander-Reiten triangles, and ℛ a functorially finite rigid subcategory of 𝒞. It is well known that there exist Auslander-Reiten sequences in mod ℛ. In this paper, we give explicitly the relations between the Auslander-Reiten translations, sequences in mod ℛ and the Auslander-Reiten functors, triangles in 𝒞, respectively. Furthermore, if 𝒯 is a cluster-tilting subcategory of 𝒞 and mod 𝒯 is a Frobenius category, we also get the Auslander-Reiten functor and the translation functor of mod 𝒯 corresponding to the ones in 𝒞. As a consequence, we get that if the quotient of a d-Calabi-Yau triangulated category modulo a cluster tilting subcategory is Frobenius, then its stable category is (2d-1)-Calabi-Yau. This result was first proved by Keller and Reiten in the case d=2, and then by Dugas in the general case, using different methods.