Nakayama Algebras which are Higher Auslander Algebras
Emre Sen
备注：
Comments are welcome! v.4 Major revision according to referee report. Main results are same but proofs are elaborated. For linear case, we add new section. v.3 We change the term repetition with covering. CM. Ringel pointed out a bug in thm 1.7. Now, it is fixed by introducing new concept "Nakayama cycle" which is suggested by CM. Ringel. We add more examples
We prove that any cyclic Nakayama algebra which is a higher Auslander algebra can be uniquely constructed from Nakayama algebras of smaller ranks by reversing the syzygy filtration process. This creates chains of higher Auslander algebras upto $\boldsymbol\varepsilon$-equivalences. Therefore, the classification of all cyclic Nakayama algebras which are higher Auslander algebras reduces to the classification of linear ones. We give two applications of this: for any integer $k$ where $2\leq k\leq 2n-2$, there is a Nakayama algebra of rank $n$ which is a higher Auslander algebra of global dimension $k$ and the possible values of the global dimensions of cyclic Nakayama algebras which are higher Auslander algebras form the sets $\left\{2,\ldots,2n-2\right\}\setminus\left\{n-1\right\}$ if $n$ is even and $\left\{2,\ldots,2n-2\right\}\setminus\left\{ 2,n-1\right\}$ if $n$ is odd.
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}.
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.
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 introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander--Reiten theory. More precisely, for each Nakayama algebra $A$ and each positive integer $d$, we construct a finite dimensional algebra $A^{(d)}$ having a distinguished $d$-cluster-tilting $A^{(d)}$-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of $A$. We also construct higher dimensional analogues of the mesh category of type $\mathbb{ZA}_\infty$ and the tubes.
We give a simple proof, using Auslander-Reiten theory, that the preprojective algebra of a basic hereditary algebra of finite representation type is Frobenius. We then describe its Nakayama automorphism, which is induced by the Nakayama functor on the module category of our hereditary algebra.
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…
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.
$n\mathbb{Z}$-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…
In this paper, we introduce the notion of excellent extension of rings. Let $Γ$ be an excellent extension of an artin algebra $Λ$, we prove that $Λ$ satisfies the Gorenstein symmetry conjecture (resp. finitistic dimension conjecture, Auslander-Gorenstein conjecture, Nakayama conjecture) if and only if so does $Γ$. As a special case of excellent extensions, if $G$ is a finite group whose order is invertible in $Λ$ acting on $Λ$ and $Λ$ is $G$-stable, we prove that if the skew group algebras $ΛG$ satisfies strong Nakayama conjecture (resp. generalized Nakayama conjecture), then so does $Λ$.
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 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 $\…
In this paper we study right $n$-Nakayama algebras. Right $n$-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra $Λ$ is representation-finite if and only if $Λ$ is right $n$-Nakayama for some positive integer $n$. We classify hereditary right $n$-Nakayama algebras. We also define right $n$-coNakayama algebras and show that an artin algebra $Λ$ is right $n$-coNakayama if and only if $Λ$ is left $n$-Nakayama. We then study right $2$-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences ove…
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 give another proof of the recent result of Ringel, which asserts equality between the finitistic dimension and delooping level of Nakayama algebras. The main tool is syzygy filtration method introduced in \cite{sen2019}. In particular, we give characterization of the finitistic dimension one Nakayama algebras.
We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.
Optimal upper bounds are provided for the dominant dimensions of Nakayama algebras and more generally algebras $A$ with an idempotent $e$ such that there is a minimal faithful injective-projective module $eA$ and such that $eAe$ is a Nakayama algebra. This answers a question of Abrar and proves a conjecture of Yamagata for monomial algebras.
The magnitude for algebras is a generalization of the Euler characteristic. We investigate the magnitude for Nakayama algebras. Using Ringel's resolution quiver, the existence and the value of rational magnitude is given. As a result, we show that the finite global dimension criteria for Nakayama algebras of Madsen and the first author are equivalent.
We review the theory of Co-Gorenstein algebras, which was introduced by Beligiannis in the article "The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, Auslander-Buchweitz Contexts and (Co-)Stabilization". We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture.
We describe the combinatorics of the cell structure of the tensor category of bimodules over a radical square zero Nakayama algebra. This accounts to an explicit description of left, right, and two-sided cells.
Using the resolution quiver for a connected Nakayama algebra, a fast algorithm is given to decide whether its global dimension is finite or not and whether it is Gorenstein or not. The latter strengthens a result of Ringel.
Let $A$ be a Nakayama algebra with $n$ simple modules and a simple module $S$ of even projective dimension $m$. Choose $m$ minimal such that a simple $A$-module with projective dimension $2m$ exists, then we show that the global dimension of $A$ is bounded by $n+m-1$. This gives a combined generalisation of results of Gustafson \cite{Gus} and Madsen \cite{Mad}. In \cite{Bro}, Brown proved that the global dimension of quasi-hereditary Nakayama algebras with $n$ simple modules is bounded by $n$. Using our result on the bounds of global dimensions of Nakayama algebras, we give a short new proof o…
Recently, we obtained in [7] a new characterization for an orthogonal system to be a simple-minded system in the stable module category of any representation-finite self-injective algebra. In this paper, we apply this result to give an explicit construction of simple-minded systems over self-injective Nakayama algebras.
If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's resolution quiver. In [IZ92], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example tha…
Let $A$ be a Nakayama algebra. We give a description of the singularity category of $A$ inside its stable module category. We prove that there is a duality between the singularity category of $A$ and the singularity category of its opposite algebra. As a consequence, the resolution quiver of $A$ and the resolution quiver of its opposite algebra have the same number of cycles and the same number of cyclic vertices.
Enomoto showed for finite dimensional algebras that the classification of exact structures on the category of finitely generated projective modules can be reduced to the classification of 2-regular simple modules. In this article, we give a combinatorial classification of 2-regular simple modules for Nakayama algebras and we use this classification to answer several natural questions such as when there is a unique exact structure on the category of finitely generated projective modules for Nakayama algebras. We also classify 1-regular simple modules, quasi-hereditary Nakayama algebras and Naka…
K. Igusa and G. Todorov introduced the $φ$ function which generalizes the notion of projective dimension. We study the behavior of the $φ$ function for cyclic Nakayama algebras of infinite global dimension. We prove that the supremum of values of $φ$ is always an even number. In particular we show that the $φ$-dimension is $2$ if and only if the algebra satisfies certain symmetry conditions. Also we give a sharp upper bound for $φ$-dimension in terms of the number of monomial relations which describes the algebra.
We study the relation between simple-minded systems and two-term tilting complexes for self-injective Nakayama algebras. More precisely, we show that any simple-minded system of a self-injective Nakayama algebra is the image of the set of simple modules under a stable equivalence, which is given by the restriction of a standard derived equivalence induced by a two-term tilting complex. We achieve this by exploiting and connecting the mutation theories from the combinatorics of Brauer tree, configurations of stable translations quivers of type A, and triangulations of a punctured convex regular…
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.
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.
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…
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.
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.
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.
