A subcategory $\mathscr{W}$ of an abelian category is called wide if it is closed under kernels, cokernels, and extensions. Wide subcategories are of interest in representation theory because of their links to other homological and combinatorial objects, established among others by Ingalls-Thomas and Marks-Šťovíček.
If $d \geqslant 1$ is an integer, then Jasso introduced the notion of $d$-abelian categories, where kernels, cokernels, and extensions have been replaced by longer complexes. Wide subcategories can be generalised to this situation.
Important examples of $d$-abelian categories arise as the $d$-cluster tilting subcategories $\mathscr{M}_{n,d}$ of $\operatorname{mod} A_n^{d-1}$, where $A_n^{d-1}$ is a higher Auslander algebra of type $A$ in the sense of Iyama. This paper gives a combinatorial description of the wide subcategories of $\mathscr{M}_{n,d}$ in terms of what we call non-interlacing collections.
This paper studies the existence of Auslander-Reiten sequences in subcategories of mod A, where A is a finite dimensional algebra over a field. The two main theorems give necessary and sufficient conditions for the existence of Auslander-Reiten sequences in subcategories. Let M be a subcategory of mod A closed under extensions and direct summands, and let X be an indecomposable module in M such that Ext^{1}(X, X') is not zero for some X' in M. Then the following are equivalent: (i) DTrX has a precover in the stable category of mod A, (ii) There exists an Auslander-Reiten sequence 0 --> U --> W…
A relative version of Auslander's formula with respect to a contravariantly finite subcategory will be given. Dual version will be treated. Several examples and applications will be provided. In particular, we show that under certain circumstances, if relative Auslander algebras of artin algebras $Λ$ and $Λ'$ are Morita equivalent, then $Λ$ and $Λ'$ are also Morita equivalent.
Auslander-Reiten duality for module categories is generalized to some sufficiently nice subcategories. In particular, our consideration works for $\mathcal{P}^{<\infty}(Λ)$, the subcategory consisting of finitely generated modules with finite projective dimension over an artin algebra $Λ$, and also, the subcategory of Gorenstein projectove modules of $\rm{mod}\mbox{-}Λ$, denoted by $\rm{Gprj}\mbox{-}Λ$. In this paper, we give a method to compute the Auslander-Reiten translation in $\mathcal{P}^{<\infty}(Λ)$ whenever $Λ$ is a $1$-Gorenstein algebra. In addition, we characterize when the Ausland…
We analyze Auslander-Reiten quivers of functorially finite resolving subcategories. Chapter 1 gives a short introduction into the basic definitions and theorems of Auslander-Reiten theory in A-mod. We generalize these definitions and theorems in Chapter 2 and prove generalizations of the first and one and a half Brauer-Thrall conjecture for functorially finite resolving subcategories. Moreover, we show that sectional paths in Auslander-Reiten-quivers are invariants of decompositions of morphisms into sums of compositions of irreducible morphisms between indecomposable modules and are strongly …
This paper introduces Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem of the paper shows that there is a close connection with covers and envelopes, also known as minimal right- and left-approximations. Namely, under suitable assumptions, if M is an object in the subcategory C of the triangulated category T and X --> Y --> M --> is an Auslander-Reiten triangle in T, then there is an Auslander-Reiten triangle K --> L --> M --> in C if and only if there is a C-cover of the form K --> X. The main theorem is used to give a new proof of the existence of Ausl…
In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from eac…
Let $Φ$ be a finite dimensional algebra over an algebraically closed field $k$ and assume gldim$\,Φ\leq d$, for some fixed positive integer $d$. For $d=1$, Brüning proved that there is a bijection between the wide subcategories of the abelian category mod$\,Φ$ and those of the triangulated category $\mathcal{D}^b(\text{mod}Φ)$. Moreover, for a suitable triangulated category $\mathcal{M}$, Jørgensen gave a description of Auslander-Reiten triangles in the extension closed subcategories of $\mathcal{M}$. In this paper, we generalise these results for $d$-abelian and $(d+2)$-angulated categories, …
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 Smalø. 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 esta…
Let $Λ$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $Λ$ admits a trivial maximal 1-orthogonal subcategory of $\modΛ$, then for any indecomposable module $M \in \mod Λ$, we have that the projective dimension of $M$ is equal to one if and only if so is its injective dimension and that $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $Λ$ is a tilted algebra.
We generalize the notions of $n$-cluster tilting subcategories and $τ$-selfinjective algebras into $n$-precluster tilting subcategories and $τ_n$-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 (Auslander--Solberg, 1993) as well as a Gorenstein analog of $n$-Auslander correspondence (Iyama,…
Auslander-Reiten theory is fundamental to study categories which appear in representation theory, for example, modules over artin algebras, Cohen-Macaulay modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves on projective curves. In these Auslander-Reiten theories, the number `2' is quite symbolic. For one thing, almost split sequences give minimal projective resolutions of simple functors of projective dimension `2'. For another, Cohen-Macaulay rings of Krull-dimension `2' provide us with one of the most beautiful situation in representation theory, which is closely r…
Let $Φ$ be a finite dimensional algebra over a field $k$. Kleiner described the Auslander-Reiten sequences in a precovering extension closed subcategory $\mathcal{X}\subseteq$ mod $Φ$. If $X\in\mathcal{X}$ is an indecomposable such that Ext$_Φ^1(X,\mathcal{X})\neq 0$ and $ζX$ is the unique indecomposable direct summand of the $\mathcal{X}$-cover $g:Y\rightarrow D$Tr$X$ such that Ext$_Φ^1(X,ζX)\neq 0$, then there is an Auslander-Reiten sequence in $\mathcal{X}$ of the form \begin{align*} ε: 0\rightarrow ζX\rightarrow X'\rightarrow X\rightarrow 0. \end{align*} Moreover, when End$_Φ(X)$ modulo th…
Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified version of higher Auslander correspondence in the context of exact categories. We define n-Auslander exact categories and show that there is a bijection between the equivalence classes of n-cluster tilting subcategories of exact categories and the equivalence classes of n-Auslander exact categories.
We give a characterization of $n$-cluster tilting subcategories of representation-directed algebras based on the $n$-Auslander-Reiten translations. As an application we classify acyclic Nakayama algebras with homogeneous relations which admit an $n$-cluster tilting subcategory. Finally, we classify Nakayama algebras of global dimension $d<\infty$ which admit a $d$-cluster tilting subcategory.
The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor $τ_n$ plays an important role in the study of $n$-cluster tilting subcategories. We study the category $\MM_n$ of preinjective-like modules obtained by applying $τ_n$ to injective modules repeatedly. We call a finite dimensional algebra $Λ$ \emph{$n$-complete} if $\MM_n=\add M$ for an $n$-cluster tilting object $M$. Our main result asserts that the endomorphism algebra $\End_Λ(M)$ is $(n+1)$-comp…
For a path algebra $A$ over a quiver $Q$, there are bijections between the support-tilting modules of $A$, torsion classes in $\mathrm{mod}(A)$ and wide subcategories in $\mathrm{mod}(A)$; these are part of the Ingalls-Thomas bijections. As a blueprint for further study, we show how these bijections manifest themselves for higher Auslander algebras of linearly oriented type $A$. In particular, we introduce a higher analogue of torsion classes in $d$-representation-finite algebras.
Auslander and Ringel-Tachikawa have shown that for an artinian ring R of finite representation type, every R-module is the direct sum of finitely generated indecomposable R-modules. In this paper, we will adapt this result to finite representation type full subcategories of the module category of an artinian ring which are closed under subobjects and direct sums and contain all projective modules. In particular, the results in this paper hold for subspace representations of a poset, in case this subcategory is of finite representation type.
For an Artinian $(n-1)$-Auslander algebra $Λ$ with global dimension $n(\geq 2)$, we show that if $Λ$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\modΛ$, then $Λ$ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in $\modΛ$ 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 $\modΛ$, 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 …
This paper has been withdrawn by the authors because it has been combined with "Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories" (arXiv:0903.0761) together. Please see the new version of the latter paper for the details.
We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $Λ$ satisfies the Auslander condition (that is, $Λ$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion pairs.
We study Auslander correspondence from the viewpoint of higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories. We give homological characterizations of Auslander algebras, especially an answer to a question of M. Artin. They are also closely related to Auslander's representation dimension of artin algebras and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities.
We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will study a chain of rejective subcategories to construct modules with endomorphisms rings of finite global dimension. We apply these result to study a function $r_Λ:\modΛ\to\nnn_{\ge0}$ which is a natural extension of Auslander's representation dimension.
We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce $n$-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an ${\rm Ext}$-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we define two additive subcategories $\mathscr{C}_r$ and $\mathscr{C}_l$ of $\mathscr{C}$ in terms of the representable functors from the stable category of $\mathscr{C}$ 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 …
The homological theory of Auslander-Platzeck-Todorov on idempotent ideals laid much of the groundwork for higher Auslander-Reiten theory, providing the key technical lemmas for both higher Auslander correspondence as well as the construction of higher Nakayama algebras, among other results. Given a finite-dimensional algebra $A$ and idempotent $e$, we expand on a criterion of Jasso-Külshammer in order to determine a correspondence between the $2$-precluster-tilting subcategories of $\mathrm{mod}(A)$ and $\mathrm{mod}(A/\langle e\rangle)$. This is then applied in the context of generalising dim…
We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander-Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander-Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes and quasi-coherent sheaves, but also to characterize so…
$n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher dimensional Auslander-Reiten theory. We give a complete classification of $n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama algebras. In particular, we show that there are three kinds of Nakayama algebras that admit $n\mathbb{Z}$-cluster tilting subcategories: finite global dimension, selfinjective and non-Iwanaga-Gorenstein. Only the selfinjective ones can admit more than one $n\mathbb{Z}$-cluster tilting subcategory. It has been shown by the second author, that each such $n\mathbb{Z}$-clust…
Spherical modules and the Auslander--Gorenstein condition for Auslander--Yoneda algebras
Mathematical Institute of the University of Bonn（波恩大学数学研究所）；Faculty of Mathematics, Kyushu University（九州大学理学研究院数学系）；Mathematisches Institut, Universität zu Köln（科隆大学数学研究所）
We classify certain resolving subcategories of finitely generated modules over a commutative noetherian ring R by using integer-valued functions on Spec R. As an application we give a complete classification of resolving subcategories when R is a locally hypersurface ring. Our results also recover a "missing theorem" by Auslander.
This paper investigates a certain 2-Calabi-Yau triangulated category D whose Auslander-Reiten quiver is ZA_{\infty}. We show that the cluster tilting subcategories of D form a so-called cluster structure, and we classify these subcategories in terms of what one may call `triangulations of the infinity-gon'. This is reminiscent of the cluster category C of type A_n which is a 2-Calabi-Yau triangulated category whose Auslander-Reiten quiver is a quotient of ZA_n. The cluster tilting subcategories of C form a cluster structure and they are classified in terms of triangulations of the (n+3)-gon. T…
Ringel's right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline-Parshall-Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra $A$ is strongly quasi-hereditary if and only if $A$ is a Nakayama algebra.
For a suitable triangulated category $\mathcal{T}$ with a Serre functor $S$ and a full precovering subcategory $\mathcal{C}$ closed under summands and extensions, an indecomposable object $C$ in $\mathcal{C}$ is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there is no Auslander-Reiten triangle in $\mathcal{C}$ with end term $C$. In this paper, we show that if, for such an object $C$, there is a minimal right almost split morphism $β:B\rightarrow C$ in $\mathcal{C}$, then $C$ appears in something very similar to an Auslander-Reiten triangle in $\mathcal{C}$: an essentially unique tri…
In 1996, Doty, Nakano and Peters defined infinitesimal Schur algebras, combining the approach via polynomial representations with the approach via $G_r T$-modules to representations of the algebraic group $G = \mathrm{GL}_n$. We study analogues of these algebras and their Auslander-Reiten theory for reductive algebraic groups $G$ and Borel subgroups $B$ by considering the categories of polynomial representations of $G_r T$ and $B_r T$ as full subcategories of $\mathrm{mod} \thinspace G_r T$ and $\mathrm{mod}\thinspace B_r T$, respectively. We show that every component $Θ$ of the stable Ausland…
From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $Λ$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$Λ$. We show that $\mathcal{M}$ is pure semisimple if and only if each module in $\mathcal{M}$ is a direct sum of finitely generated modules. Let $\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$Λ$. We show that $Add(\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$Λ$ if and only if $\mathfrak{m}$ has an additive generator…
Let $\mathbf{k}$ be an algebraically closed field, let $Λ$ be a finite dimensional $\mathbf{k}$-algebra, and let $\widehatΛ$ be the repetitive algebra of $Λ$. For the stable category of finitely generated left $\widehatΛ$-modules $\widehatΛ$-\underline{mod}, we show that the irreducible morphisms 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. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $\widehatΛ$-\underline{mod}. We…
For an $(n-1)$-Auslander algebra $Λ$ with global dimension $n$, we give some necessary conditions for $Λ$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $Λ$-modules with projective dimension $n-1$ or $n$. For an almost hereditary algebra $Λ$ with global dimension 2, we prove that $Λ$ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable $Λ$-module $M$, $M$ is injective is equivalent to that the reduced grade of $M$ is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence o…
A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If $Φ$ is a finite dimensional algebra, then each functorially finite wide subcategory of $\operatorname{mod}( Φ)$ is of the form $ϕ_{ * }\big( \operatorname{mod}( Γ) \big)$ in an essentially unique way, where $Γ$ is a finite dimensional algebra and $Φ\stackrel{ ϕ}{ \longrightarrow } Γ$ is an algebra epimorphism satisfying $\operatorname{Tor}^{ Φ}_1( Γ,Γ) = 0$. Let ${\mathcal F}…
In this paper, we prove a reduction result on wide subcategories of abelian categories which is similar to Calabi-Yau reduction, silting reduction and $τ$-tilting reduction. More precisely, if an abelian category $\mathcal{A}$ admits a recollement relative to abelian categories $\mathcal{A}'$ and $\mathcal{A}"$, diagrammatically expressed by $$\xymatrix@!C=2pc{ \mathcal{A'} \ar@{>->}[rr]|{i_{*}} && \mathcal{A} \ar@<-4.0mm>@{->>}[ll]_{i^{*}} \ar@{->>}[rr]|{j^{*}} \ar@{->>}@<4.0mm>[ll]^{i^{!}}&& \mathcal{A''} \ar@{>->}@<-4.0mm>[ll]_{j_{!}} \ar@{>->}@<4.0mm>[ll]^{j_{*}} },$$ then the assignment $…
In a paper by Holm and Jorgensen, the cluster category $\mathscr{D}$ of type $A_\infty$, with Auslander-Reiten quiver $\mathbb{Z} A_\infty$, is introduced. Slices in the Auslander-Reiten quiver of $\mathscr{D}$ give rise to direct systems; the homotopy colimit of such direct systems can be computed and these "Prüfer objects" can be adjoined to form a larger category. It is this larger category, $\overline{\mathscr{D}},$ which is the main object of study in this paper. We show that $\overline{\mathscr{D}}$ inherits a nice geometrical structure from $\mathscr{D}$; "arcs" between non-neighbouring…
For an indecomposable module $M$ over a path algebra of a quiver of type $\widetilde{\mathbb A}_n$, the Gabriel-Roiter measure gives rise to four new numerical invariants; we call them the multiplicity, and the initial, periodic and final parts. We describe how these invariants for $M$ and for its dual specify the position of $M$ in the Auslander-Reiten quiver of the algebra.
We apply the notion of a full convex subcategory to a wide range of algebras including tilted, quasi-tilted, shod, weakly shod, left and right glued, laura, simply connected, strongly simply connected, left supported, and cluster-tilted. In particular, given an algebra $Λ$ from one of the aforementioned classes, we investigate certain factor algebras $Λ/I$ where $I$ is an ideal generated by a suitable idempotent.
In this paper, we study representations of certain string algebras, which are referred to as of affine type $\widetilde{C}$. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and $τ$-locally free modules defined by Geiss-Lerclerc-Schröer. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type $\widetilde{C}$ and $τ$-locally free modules of the corresponding string algebras.
An algebra is said to be \emph{$τ$-tilting finite} provided it has only a finite number of $τ$-rigid objects up to isomorphism. We associate a category to each such algebra. The objects are the wide subcategories of its category of finite dimensional modules, and the morphisms are indexed by support $τ$-tilting pairs.
