The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $\mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $\mathcal{E}$ are reflected in $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ as a subcategory of $\operatorname{mod}(\mathcal{E})$ when $\mathcal{E}$ 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 $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\mathcal{C}$ and certain resolving subcategories of $\operatorname{mod}(\mathcal{C})$.
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.
A celebrated result in representation theory is that of higher Auslander correspondence. Let $Λ$ an Artin algebra and $X$ a $d$-cluster-tilting module. Iyama has shown that the endomorphism ring $Γ$ of $X$ is a $d$-Auslander algebra, and moreover this gives a correspondence between $d$-cluster-tilting modules and $d$-Auslander algebras. We present a self-contained and concise proof using the homological theory of idempotent ideals of Auslander--Platzeck--Todorov.
Let $A$ be an artin algebra. We show that the bounded homotopy category of finitely generated right $A$-modules has Auslander-Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [H2]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules admits Auslander-Reiten triangles, which improves a main result in [G].
Quasi-hereditary were introduced by L. Scott \cite{Scott, CPS1,CPS2} in order to deal highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups, and they have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander [A], [AQM] and used in his proof of the first Brauer-Thrall conjecture [A2] and later on used systematically in his joint work with I. Reiten on stable equivalence [AR], [AR2] and many other applications. R…
It is shown that a 4-recollement of derived categories of CM-finite algebras induces a 2-recollement of the corresponding Cohen-Macaulay Auslander algebras, which generalises the main theorem of Pan [S. Y. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra 216 (2012), 355{363]. Moreover, both Auslander- Reiten conjecture and Gorenstein projective conjecture are shown invariant under 3 (or 4)-recollement of unbounded derived categories of algebras.
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 give analogues of the Auslander correspondence for two classes of triangulated categories satisfying certain finiteness conditions. The first class is triangulated categories with additive generators and we consider their endomorphism algebras as the Auslander algebras. For the second one, we introduce the notion of $[1]$-additive generators and consider their graded endormorphism algebras as the Auslander algebras. We give a homological characterization of the Auslander algebras for each class. Along the way, we also show that the algebraic triangle structures on the homotopy categories ar…
We generalize Ringel and Schmidmeier's theory on the Auslander-Reiten translation of the submodule category $\mathcal S_2(A)$ to the monomorphism category $\mathcal S_n(A)$. As in the case of $n=2$, $\mathcal S_n(A)$ has Auslander-Reiten sequences, and the Auslander-Reiten translation $τ_{\mathcal{S}}$ of $\mathcal S_n(A)$ can be explicitly formulated via $τ$ of $A$-mod. Furthermore, if $A$ is a selfinjective algebra, we study the periodicity of $τ_{\mathcal{S}}$ on the objects of $\mathcal S_n(A)$, and of the Serre functor $F_{\mathcal S}$ on the objects of the stable monomorphism category $\…
Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a $\mathbb{D}^{\rm{b}}({\rm{mod\mbox{-}}} )$ level recollement, writing $A$ in terms of $B$ and $C$, then we get a $\mathbb{D}^-({\rm{Mod\mbox{-}}} )$ level recollement of certain functor categories, induces from the module categories of $A$, $B$ and $C$. As an application, we generalise the main theorem of Pan [Sh. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra, 216 (2012), 355-363] in terms of recollements of Gorenstein artin algebras. Moreover, we…
According to the Auslander's formula one way of studying an abelian category ${\mathcal{C}}$ is to study ${\rm mod}\mbox{-}{\mathcal{C}}$, that has nicer homological properties than ${\mathcal{C}}$, and then translate the results back to ${\mathcal{C}}$. Recently Krause gave a derived version of this formula and thus renewed the subject. This paper contains a detailed study of various versions of Auslander formula including the versions for all modules and for unbounded derived categories. We apply them to include some results concerning recollements of triangulated categories.
Let $\cal C$ be a $(d+2)$-angulated category. In this note, we show that if $\cal C$ is a locally finite, then $\cal C$ has Auslander-Reiten $(d+2)$-angles. This extends a result of Xiao-Zhu for triangulated categories.
We study the Auslander ring of a basic left Köthe ring $Λ$ and give a characterization of basic left Köthe rings in terms of their Auslander rings. We also study the functor category Mod$((Λ$-mod$)^{\rm op})$ and characterize basic left Köthe rings $Λ$ by using functor categories Mod$((Λ$-mod$)^{\rm op})$. As a consequence we show that there exists a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings.
Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we prove that $\mathscr{C}$ has Auslander-Reiten-Serre duality if and only if $\mathscr{C}$ has Auslander-Reiten $n$-exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander-Reiten-Serre duality). Finally, assume further that $\mathscr{C}$ has Auslander-Reiten-Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection …
Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.
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…
Let $R$ be a commutative artinian ring. We extend higher Auslander correspondence from Artin $R$-algebras of finite representation type to dualizing $R$-varieties. More precisely, for a positive integer $d$, we show that a dualizing $R$-variety is $d$-abelian if and only if it is a $d$-Auslander dualizing $R$-variety if and only if it is equivalent to a $d$-cluster-tilting subcategory of the category of finitely presented modules over a dualizing $R$-variety.
For a finite group $G$ and an algebraically closed field $k$ of characteristic $p>0$ for any indecomposable finite dimensional $kG$-module $M$ with vertex $D$ and a subgroup $H$ of $G$ containing $N_G(D)$ there is a unique indecomposable $kH$-module $N$ of vertex $D$ being a direct summand of the restriction of $M$ to $H$. This correspondence, called Green correspondence, was generalised by Auslander-Kleiner to the situation of pairs of adjoint functors between additive categories. In the original situation of group rings Carlson-Peng-Wheeler proved that this correspondence is actually restric…
Let $\mathcal{A}$ be a locally bounded $k$-category and $G$ a torsion-free group of $k$-linear automorphisms of $\mathcal{A}$ acting freely on the objects of $\mathcal{A},$ and $F:\mathcal{A}\rightarrow \mathcal{B}$ is a Galois functor. We extend naturally the push-down functor $F_λ$ to the functor $\rm{H}\rm{F}_λ:\rm{H}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S} \rm{F}_λ:\mathcal{S}(\rm{mod}\mbox{-} \mathcal{A})\rightarrow \mathcal{S}(\rm{mod}\mbox{-} \mathcal{B})$, between the corresponding morphism categories, resp. monomorphism catego…
Let $Λ$ be an artin algebra and $S(Λ)$ the category of all embeddings $(A\subseteq B)$ where $B$ is a finitely generated $Λ$-module and $A$ is a submodule of $B$. Then $S(Λ)$ is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in $S(Λ)$ can be computed within the category of $Λ$-modules by using our construction of minimal monomorphisms. If in addition $Λ$ is uniserial then any nonprojective indecomposable object in $\Cal S(Λ)$ is invariant under the sixth power of the Auslander-Reiten translation.
We construct two functors from the submodule category of a self-injective representation-finite algebra $Λ$ to the module category of the stable Auslander algebra of $Λ$. These functors factor through the module category of the Auslander algebra of $Λ$. Moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism. Their construction uses a recollement of the module category of the Auslander algebra induced by an idempotent and this recollement determines a ch…
The Caldero-Chapoton formula relates for hereditary algebras of Dynkin type the cluster characters of the end terms of an Auslander-Reiten sequence with the cluster character of the middle term. We extend this result to generalized cluster categories with cluster tilting object by considering Auslander-Reiten triangles.
Let $\mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $\mathscr{C}$ is locally finite, then $\mathscr{C}$ has Auslander-Reiten $n$-exangles. This unifies and extends results of Xiao-Zhu, Zhu-Zhuang, Zhou and Xie-Lu-Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively.
We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander-Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.
Let $\text{Gprj}\mbox{-}Λ$ denote the category of Gorenstein projective modules over an Artin algebra $Λ$ and the category $\text{mod}\mbox{-} (\underline{\text{Gprj}}\mbox{-}Λ)$ of finitely presented functors over the stable category $\underline{\text{Gprj}}\mbox{-}Λ$. In this paper, we study those algebras $Λ$ with $\text{mod}\mbox{-} (\underline{\text{Gprj}}\mbox{-}Λ)$ to be a semisimple abelian category, and called $Ω_{\mathcal{G}}$-algebras. The class of $Ω_{\mathcal{G}}$-algebras contains important classes of algebras, including gentle algebras. Over an $Ω_{\mathcal{G}}$-algebra $Λ$, the…
Let $Λ$ be an Artin algebra and let $\rm{Gprj}\mbox{-}Λ$ denote the class of all finitely generated Gorenstein projective $Λ$-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category $\mathcal{S}({\rm Gprj}\mbox{-}Λ)$ containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category $\underline{\rm{Gprj}}\mbox{-}Λ$. In particular, for the finite components, we show that under certain mild conditions t…
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 midterms of almost split sequences. As such, another part of th…
Let $\mathscr{C}$ be a $(d+2)$-angulated category with $d$-suspension functor $Σ^d$. Our main results show that every Serre functor on $\mathscr{C}$ is a $(d+2)$-angulated functor. We also show that $\mathscr{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander--Reiten $(d+2)$-angles. Moreover, $τ_d=\mathbb{S}Σ^{-d}$ where $τ_d$ is $d$-Auslander-Reiten translation. These results generalize work by Bondal-Kapranov and Reiten-Van den Bergh. As an application, we prove that for a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$, the quotient categ…
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…
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 $(\mathscr{C},\mathbb{E},\mathfrak{s})$ 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 …
We revisit faithfully balanced modules. These are faithful modules having the double centralizer property. For finite-dimensional algebras our main tool is the category ${\rm cogen}^1(M)$ of modules with a copresentation by summands of finite sums of $M$ on which ${\rm Hom}(-,M)$ is exact. For a faithfully balanced module $M$ the functor ${\rm Hom}(-,M)$ is a duality on these categories - for cotilting modules this is the Brenner-Butler theorem. We also study new classes of faithfully balanced modules combining cogenerators and cotilting modules. Then we turn to relative homological algebra in…
It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409.8611 that for punctured surfaces $X$ with stops $Λ$ at their boundary, the partially wrapped Fukaya category $\mathcal{W}(X,Λ)$ provides a categorical (smooth and proper) resolution of the compact Fukaya category $\mathcal{F}(X)$. Inspired by thi…
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…
We introduce a notion of generalized Auslander-Reiten duality on a Hom-finite Krull-Schmidt exact category $\mathcal{C}$. This duality induces the generalized Auslander-Reiten translation functors $τ$ and $τ^-$. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories $\mathcal{C}_r$ and $\mathcal{C}_l$ of $\mathcal{C}$. A non-projective indecomposable object lies in the domain of $τ$ 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 $τ^-$ if and only if it a…
Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $Δ_R(A)$ of $A$ has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category $D_{sg}(R)$ of Buchweitz and Orlov as a certain canonical quotient category. If $R$ has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that $D_{sg}(R)$ determines $Δ_R(Aus(R))$, where $Aus(R)$ is the corresponding Auslander algebra. …
Xiao and Zhu has shown that if $\cal C$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\cal C$. The notion of $(d +2)$-angulated categories is a "higher dimensional" analogue of triangulated categories. In this article, we show that if A $(d+2)$-angulated category $\cal C$ is locally finite if and only if the Auslander-Reiten $(d+2)$-angles generate the relations for the Grothendieck group of $\cal C$. This extends the result of Xiao and Zhu, and gives the converse of Xiao and Zhu's result is also true.
We analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quiver is determined. We also determine components that contain shift periodic complexes.
Let $\La$ be an elementary locally bounded linear category over a field with radical squared zero. We shall show that the bounded derived category $D^b(\ModbLa)$ of finitely supported left $\La$-modules admits a Galois covering which is the bounded derived category of almost finitely co-presented representations of a gradable quiver. Restricting to the bounded derived category $D^b({\rm mod}^b\hspace{-2pt}\La)$ of finite dimensional left $\La$-modules, we shall be able to describe its indecomposable objects, obtain a complete description of the shapes of its Auslander-Reiten components, and cl…
