Let $Λ$ be a finite dimensional Auslander algebra. For a $Λ$-module $M$, we prove that the projective dimension of $M$ is at most one if and only if the projective dimension of its socle soc\,$M$ is at most one. As an application, we give a new characterization of Auslander algebra $Λ$, and prove that a finite dimensional algebra $Λ$ is an Auslander algebra provided its global dimension gl.d\,$Λ\leq2$ and an injective $Λ$-module is projective if and only if the projective dimension of its socle is at most one.
In this article we investigate the relations between the Gorenstein projective dimensions of $Λ$-modules and their socles for minimal n-Auslander-Gorenstein algebras $Λ$ in the sense of Iyama and Solberg \cite{IS}. First we give a description of projective-injective $Λ$-modules in terms of their socles. Then we prove that a $Λ$-module $N$ has Gorenstein projective dimension at most n iff its socle has Gorenstein projective dimension at most n iff $N$ is cogenerated by a projective $Λ$-module. Furthermore, we show that minimal n-Auslander-Gorenstein algebras can be characterised by the relation…
An artin algebra $A$ is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective $A$-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type (\cite{A,A1}).
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.
The ADR algebra $R_A$ of an Artin algebra $A$ is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the $Δ$-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of $R_A$-modules. As an application, we give a counterexample to a claim by Auslander-Platzeck-Todorov, concerning projective resolutions over the ADR algebra.
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.
The Cohen-Macaulay Auslander algebra of any string algebra is explicitly constructed in this paper. Furthermore, we show that a class of special string algebras, which are called to be string algebras with G-condition, are representation-finite if and only if their Cohen-Macaulay Auslander algebras are representation-finite. Finally, the self-injective dimension of gentle algebras is characterized using their Cohen-Macaulay Auslander algebras.
A Counterexample to Left-Right Symmetry for Diagonal Auslander-Regularity
School of Mathematics, Nanjing University（南京大学数学系）；School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）；Guizhou Provincial Key Laboratory of Applied Mathematics and Computing Power & Algorithms（贵州省应用数学与计算力及算法重点实验室）
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.
Auslander-Reiten conjecture, which says that an Artin algebra does not have any non-projective generator with vanishing self-extensions in all positive degrees, is shown to be invariant under certain singular equivalences induced by adjoint pairs, which occur often in matrix algebras, recollements and change of rings. Accordingly, several reduction methods are established to study this conjecture.
We construct an isomorphism between the partially ordered set of tilting modules for the Auslander algebra of $K[x]/(x^n)$ and the interval of rational permutation braids in the braid group on $n$ strands. Hence, there are only finitely many tilting modules.
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…
In this paper, we introduce $Φ$-Auslander-Yoneda algebras in a triangulated category with $Φ$ a parameter set in $\mathbb N$, and provide a method to construct new derived equivalences between these $Φ$-Auslander-Yoneda algebras (not necessarily Artin algebras), or their quotient algebras, from a given almost $ν$-stable derived equivalence. As consequences of our method, we have: (1) Suppose that $A$ and $B$ are representation-finite, self-injective Artin algebras with $_AX$ and $_BY$ additive generators for $A$ and $B$, respectively. If $A$ and $B$ are derived-equivalent, then the $Φ$-Ausland…
Let $Λ$ be a radical square zero Nakayama algebra with $n$ simple modules and let $Γ$ be the Auslander algebra of $Λ$. Then every indecomposable direct summand of a tilting $Γ$-module is either simple or projective. Moreover, if $Λ$ is self-injective, then the number of tilting $Γ$-modules is $2^n$; otherwise, the number of tilting $Γ$-modules is $2^{n-1}$.
We provide a combinatorial algorithm for constructing the stable Auslander-Reiten component containing a given indecomposable module of a symmetric special biserial algebra using only information from its underlying Brauer graph. We also show that the structure of the Auslander-Reiten quiver is closely related to the distinct Green walks around the Brauer graph and detail the relationship between the precise shape of the stable Auslander-Reiten components for domestic Brauer graph algebras and their underlying graph. Furthermore, we show that the specific component containing a given simple or…
This paper studies self-injective algebras of polynomial growth. We prove that the derived equivalence classification of weakly symmetric algebras of domestic type coincides with the classification up to stable equivalences (of Morita type). As for weakly symmetric non-domestic algebras of polynomial growth, up to some scalar problems, the derived equivalence classification coincides with the classification up to stable equivalences of Morita type. As a consequence, we get the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between weakly symmetric algebras o…
We introduce a new class of quasi-hereditary algebras, containing in particular the Auslander-Dlab-Ringel (ADR) algebras. We show that this new class of algebras is preserved under Ringel duality, which determines in particular explicitly the Ringel dual of any ADR algebra. As a special case of our theory, it follows that, under very restrictive conditions, an ADR algebra is Ringel dual to another one. The latter provides an alternative proof for a recent result of Conde and Erdmann, and places it in a more general setting.
We provide a framework for part of the homological theory of Z-algebras and their generalizations, directed towards analogues of the Auslander-Gorenstein condition and the associated double Ext spectral sequence that are useful for enveloping algebras of Lie algebras and related rings. As an application, we prove the equidimensionality of the characteristic variety of an irreducible representation of the Z-algebra, and for related representations over quantum symplectic resolutions. In the special case of Cherednik algebras of type A, this answers a question raised by the authors.
Let $Λ$ be a radical square zero algebra of a Dynkin quiver and let $Γ$ be the Auslander algebra of $Λ$. Then the number of tilting right $Γ$-modules is $2^{m-1}$ if $Λ$ is of $A_{m}$ type for $m\geq 1$. Otherwise, the number of tilting right $Γ$-modules is $2^{m-3}\times14$ if $Λ$ is either of $D_{m}$ type for $m\geq 4$ or of $E_{m}$ type for $m=6,7,8$.
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.
It is a well-known result of Auslander and Reiten that contravariant finiteness of the class $\mathcal{P}^{\mathrm{fin}}_\infty$ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander--Reiten condition, namely contravariant finiteness of the class $\mathcal{GP}^{\mathrm{fin}}_\infty$ (of finitely gen…
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.
Infinitely Many Components in Auslander--Reiten Quivers of Representation-Infinite Algebras over Perfect Fields
School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
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,…
Recently, Chen and Koenig in \cite{CheKoe} and Iyama and Solberg in \cite{IyaSol} independently introduced and characterised algebras with dominant dimension coinciding with the Gorenstein dimension and both dimensions being larger than or equal to two. In \cite{IyaSol}, such algebras are named Auslander-Gorenstein algebras. Auslander-Gorenstein algebras generalise the well known class of higher Auslander algebras, where the dominant dimension additionally coincides with the global dimension. In this article we generalise Auslander-Gorenstein algebras further to algebras having the property th…
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 …
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…
Let k be an algebraically closed field and A a finite-dimensional k-algebra. Given an A-module M, the set G_e(M) of all submodules of M with dimension vector e is called a quiver Grassmannian. If D,Y are A-modules, then we consider Hom(D,Y) as a B-module, where B is the opposite of the endomorphism ring of D, and the Auslander varieties for A are the quiver Grassmannians of the form G_e Hom(D,Y). Quiver Grassmannians, thus also Auslander varieties are projective varieties and it is known that every projective variety occurs in this way. There is a tendency to relate this fact to the wildness o…
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…
In this paper, we compute the Hochschild and cyclic homologies of the Auslander algebras of the Taft algebras. We also describe the first Chern character for the Taft algebras and for their Auslander algebras.
A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to non-graded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing com…
We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $Γ$ of Dynkin type $A…
Our first result provides a new characterization of Auslander algebras using a property of hereditary torsion pairs. The second result shows an Auslander algebra $Λ$ is left or right glued if and only if $Λ$ is representation-finite. Finally, our third result shows the module category of any Auslander algebra contains a tilting module with a particular property, which we call the hereditary property. Applications of this property are investigated.
This article begins the study of irreducible maps involving finite-dimensional uniserial modules over finite-dimensional associative algebras. We work on the classification of irreducible maps between two uniserials over triangular algebras, and give estimates for the number of middle terms of an almost split sequence with a uniserial end term.
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…
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.
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 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.
Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent.
We describe the structure of semi-regular Auslander-Reiten components of artin algebras without external short paths in the module category. As an application we give a complete description of self-injective artin algebras whose Auslander-Reiten quiver admits a regular acyclic component without external short paths.
We give new improved bounds for the dominant dimension of Nakayama algebras and use those bounds to give a classification of Nakayama algebras with $n$ simple modules that are higher Auslander algebras with global dimension at least $n$. The classification is then used to extend the results on the inequality for the global dimension of Nakayama algebras obtained in \cite{MM}.
Given a compact p-adic Lie group we show that its distribution algebra is Frechet-Stein with Auslander regular Banach algebras. As an application, we show that nonzero coadmissible modules coming from smooth or, more general, U(g)-finite representations have maximal codimension.
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].
We classify Borel-Schur algebras having finite representation type. We also de- termine 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.
