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 categories, of $\rm{mod}\mbox{-} \mathcal{A}$ and $\rm{mod}\mbox{-} \mathcal{B}$. Under some additional conditions, we show that $\rm{H}(\rm{mod}\mbox{-}\mathcal{A})$, resp. $\mathcal{S}( \rm{mod}\mbox{-}\mathcal{A})$, is locally bounded if and only if $\rm{H}(\rm{mod}\mbox{-} \mathcal{B})$, resp. $\mathcal{S}(\rm{mod}\mbox{-}\mathcal{B})$, is of finite representation type. As an application, we show that the stable Auslander algebra of a representation-finite selfinjective algebra $Λ$ is again representation-finite if and only if $Λ$ is of Dynkin type $\mathbb{A}_{n}$ with $n\leqslant 4$.
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…
Given an exact category $\mathcal{C}$, we denote by $\mathcal{C}_l$ the smallest additive subcategory containing injectives and indecomposable objects which appear as the first term of an almost split conflation. We prove that a deflation is right determined by some object if and only if its intrinsic kernel lies in $\mathcal{C}_l$. We give characterizations for $\mathcal{C}$ having Auslander-Reiten duality.
Let A be an artin algebra. In his seminal Philadelphia Notes published in 1978, M. Auslander introduced the concept of morphisms being determined by modules. Auslander was very passionate about these ivestigations (they also form part of the final chapter of the Auslander-Reiten-Smaloe book and could and should be seen as its culmination), but the feedback until now seems to be somewhat meager. The theory presented by Auslander has to be considered as an exciting frame for working with the category of A-modules, incorporating all what is known about irreducible maps (the usual Auslander-Reiten…
Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for constructing the connecting component of the derived category of any tilted algebra. In particular we obtain a technique for constructing the derived category of any tilted algebra of finite representation type.
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 …
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 $(\mathscr{C},\mathbb{E},\mathfrak{s})$ 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 introduce and study a new class of morphisms which includes morphisms represented by monomorphisms in the sense of Auslander and Bridger. As an application, we give not only an extension of Kato's theorem on morphisms represented by monomorphisms, but also a common generalization of several results due to Auslander and Bridger that describe relationships between torsionfreeness and the grades of Ext modules.
Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.
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…
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 $\text{Gprj}\mbox{-}Λ$ 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…
An artin algebra A is said to be a higher Auslander algebra provided the global dimension and the dominant dimension coincide. We say that a linear Nakayama algebra is monotone, provided its Kupisch series first increases, then decreases. We are going to classify the monotone Nakayama algebras which are higher Auslander algebras. Let us stress that the classification strongly depends on the parity of the global dimension of A.
We describe a procedure for constructing morphisms in additive categories, combining Auslander's concept of a morphism determined by an object with the existence of flat covers. Also, we show how flat covers are turned into projective covers, and we interprete these constructions in terms of adjoint functors.
We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smalø-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.
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…
Cohen-Macaulay Auslander algebras are the endomorphism algebras of representation generators of the subcategory of Gorenstein projective modules over $\rm{CM}$-finite algebras. In this paper, we study Cohen-Macaulay Auslander algebras over $1$-Gorenstein algebras and $Ω_{\mathcal{G}}$-algebras. $1$-Gorenstein algebras are those of algebras with global Gorenstein projective dimension at most one and $Ω_{\mathcal{G}}$-algebras are a class of algebras introduced in this paper, including some important class of algebras for example Gentle algebras and more generally quadratic monomial algebras. It…
By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over the algebraic closure ${\bar \BF}_q$ of the finite field $\BF_q$ of $q$ elements, we establish a relation between the representation theory of $A$ over ${\bar \BF}_q$ and that of the $F$-fixed point algebra $A^F$ over $\BF_q$. More precisely, we prove that the category $\modh A^F$ of finite dimensional $A^F$-modules is equivalent to the subcategory of finite dimensional $F$-stable $A$-modules, and, when $A$ is finite dimensional, we establish a bijection between the isoclasses of indecomposable $A^F$-modules and the …
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…
We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to require that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left d…
Let $\mathcal{A}$ be a Hom-finite additive Krull-Schmidt $k$-category where $k$ is an algebraically closed field. Let ${\rm mod} \mathcal{A}$ denote the category of locally finite dimensional $\mathcal{A}$-modules, that is, the category of covariant functors $\mathcal{A} \to {\rm mod} k$. We prove that an irreducible monomorphism in ${\rm mod} \mathcal{A}$ has a finitely generated cokernel, and that an irreducible epimorphism in ${\rm mod} \mathcal{A}$ has a finitely co-generated kernel. Using this, we get that an almost split sequence in ${\rm mod} \mathcal{A}$ has to start with a finitely co…
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…
We deal with finitely generated modules over an artin algebra. In his Philadelphia Notes, M.Auslander showed that any homomorphism is right determined by a module C, but a formula for C which he wrote down has to be modified. The paper includes now complete and direct proofs of the main results concerning right determiners of morphisms. We discuss the role of indecomposable projective direct summands of a minimal right determiner and provide a detailed analysis of those morphisms which are right determined by a module without any non-zero projective direct summand.
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…
For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander-Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of $SL(2)$. Moreover, for a normal subgroup scheme $\mathcal{N}$ of a finite group scheme $\mathcal{G}$, we show that there is a connection between the ramification indices of the restriction morphism $\mathbb{P}(\mathcal{V}_{\mathcal{N}})\rightarrow\mathbb{P}(\mathcal{V}_{\mathcal{G}})$ between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander-Reiten quiver…
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.
Coherent strings of composable morphisms play an important role in various important constructions in abstract stable homotopy theory (for example algebraic K-theory or higher Toda brackets) and in the representation theory of finite dimensional algebras (as representations of Dynkin quivers of type A). In a first step we will prove a strong comparison result relating composable strings of morphisms and coherent diagrams on cubes with support on a path from the initial to the final object. We observe that both structures are equivalent (by passing to higher analogues of mesh categories) to dis…
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.
Relations between categorifications of higher-dimensional type $A$ cluster combinatorics
Universität Hamburg（汉堡大学）；Institut Camille Jordan UMR 5208, Université Jean Monnet, CNRS, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1（卡米尔·乔丹研究所，让·莫奈大学，法国国家科学研究中心，里昂中央理工学院，里昂国立应用科学学院，里昂第一大学）；Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge（剑桥大学纯数学与数学统计系，数学科学中心）
We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Nakashima Kashiwara tableaux. This yields a translation of Nakajima monomials to the Littelmann path model. Furthermore, as an application of our results we describe an insertion scheme for Nakajima monomials compatible to the insertion scheme for tableaux.
Towards the study of the representation theory of any dihedral Artin group B, we build rational morphisms from B to the group of invertible elements of the associated infinitesimal braids algebra. For this we build analogues of Drinfeld associators, in rank 2 for arbitrary dihedral symmetries of the plane.
We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $\mathcal{C}$ over a local ring $R$. If the maximal ideal of $R$ is generated by a single element, we show that any thick ideal of $\mathcal{C}$ admits an explicitely given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime ch…
