A proof of the strong no loop conjecture

The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field.     

关于有限维数猜想的一些新进展

在Artin代数的表示理论中,有一个著名的有限维数猜想：任意给定一个Artin代数,它的有限维数都是有限的．这个猜想已有45年的历史,至今悬而未决．本文主要综述它的一些历史发展情况,并介绍关于有限维数猜想的一些最新进展．     

Finitistic Dimensions of Artin Algebras with Two Simples and a Conjecture of Marczinzik

Let Λ be an Artin algebra with a unique non-injective indecomposable projective module. In this situation, Marczinzik conjectured that the dominant dimension of Λ agrees with its finitistic dimension. In this paper, we give a proof of a stronger statement. As a byproduct, we obtain excellent control over the finitistic dimensions of Artin algebras with two simples and positive dominant dimension, and also establish the Gorenstein symmetry conjecture for all algebras under consideration.

     

On Modules over Endomorphism Algebras of Maximal Rigid Objects in 2-Calabi-Yau Triangulated Categories

This note investigates the modules over the endomorphism algebras of maximal rigid objects in 2-Calabi-Yau triangulated categories. We study the possible complements for almost complete tilting modules. Combining with Happel's theorem, we show that the possible exchange sequences for tilting modules over such algebras are induced by the exchange triangles for maximal rigid objects in the corresponding 2-Calabi-Yau triangulated categories. For the modules of infinite projective dimension, we generalize a recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras.     

ARTIN RINGS WITH ALL IDEMPOTENT IDEALS PROJECTIVE

We study artin rings Λ with the property that all the idempotents two sided ideals of Λ are projective left Λ-modules. We give a characterization of these rings, and prove that their finitistic dimension is at most one. Using this result we study the Λ-modules of finite projective dimension.     

On the Quasi-Stratified Algebras of Liu and Paquette

Liu and Paquette defined a class of artin algebras, more general than the standardly stratified ones, called quasi-stratified algebras. Not only is the Cartan Determinant Conjecture (CDC) true for these algebras, so is its converse. This article shows that this class of algebras is preserved under “pruning” sources and sinks from the left quiver. It compares the classes of quasi-stratified and left serial algebras, as well as quasi-stratified and gentle algebras. Holm has shown that the CDC holds for gentle algebras; the converse is also established. It is shown when a Yamagata family of algebras of large finite global dimension yield quasi-stratified ones and constructs quasi-stratified elementary algebras from smaller ones.     

一类自入射Koszul特殊双列代数的Hochschild同调群

基于Snashall与Taillefer构造的极小投射双模分解,用组合的方法,清晰地计算出一类自入射Koszul特殊双列代数∧_N的各阶Hochschild同调群的维数,从而以计算的方式直观地表明了韩阳的猜想对这类代数∧_N成立.     

Iyama’s finiteness theorem via strongly quasi-hereditary algebras

Let Λ be an artin algebra and X a finitely generated Λ-module. Iyama has shown that there exists a module Y such that the endomorphism ring Γ of X⊕Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Γ is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length n must have global dimension at most 2n−2. We want to show that Iyama’s better bound is related to the fact that the ring Γ he constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1.     

Generalized Igusa–Todorov function and finitistic dimensions

We introduce a new function from the bounded derived category of a finite dimensional algebra over a field to the set of all natural numbers, which is a generalized version of the Igusa–Todorov function. Then we extend the results corresponding to the Igusa–Todorov function. As an application, we give a new proof of the finiteness of the finitistic dimension of special biserial algebras.     

Characterization of eventually periodic modules in the singularity categories

The singularity category of a ring makes only the modules of finite projective dimension vanish among the modules, so that the singularity category is expected to characterize a homological property of modules of infinite projective dimension. In this paper, among such modules, we deal with eventually periodic modules over a left artin ring, and, as our main result, we characterize them in terms of morphisms in the singularity category. As applications, we first prove that, for the class of finite dimensional algebras over a field, being eventually periodic is preserved under singular equivalence of Morita type with level. Moreover, we determine which finite dimensional connected Nakayama algebras are eventually periodic when the ground field is algebraically closed.     

Mutation of Auslander Generators for Selfinjective Algebras

Let Λ be an artin algebra with representation dimension equal to three and M an Auslander generator of Λ. We show how, under certain assumptions, we can mutate M to get a new Auslander generator whose endomorphism ring is derived equivalent to the endomorphism ring of M. We apply our results to selfinjective algebras with radical cube zero of infinite representation type, where we construct an infinite set of Auslander generators.     

Finiteness of representation dimension

We will show that any module over an artin algebra is a direct summand of some module whose endomorphism ring is quasi-hereditary. As a conclusion, any artin algebra has a finite representation dimension.

     

Cotorsion pairs generated by modules of bounded projective dimension

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an ℵ₀-noetherian ring Q of little finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and to Cohen Macaulay noetherian commutative rings.     

Characterizations of algebras with small homological dimensions

We introduce the class of double tilted algebras, containing the class of tilted algebras and prove various characterizations. In particular, we show that the class of double tilted algebras is the class of all artin algebras whose AR-quiver admits a faithful double section with a natural property. Moreover, we prove that the class of double tilted algebras coincides with the class of all artin algebras of global dimension three, for which every indecomposable finitely generated module has projective or injective dimension at most one. We also describe the structure of the category of finitely generated modules as well as the AR-quiver of double tilted algebras.     

Defending the negated Kaplansky conjecture

To answer in the negative a conjecture of Kaplansky, four recent papers independently constructed four families of Hopf algebras of fixed finite dimension, each of which consisted of infinitely many isomorphism classes. We defend nevertheless the negated conjecture by proving that the Hopf algebras in each family are cocycle deformations of each other.

     

On artin algebras with almost all indecomposable modules of projective or injective dimension at most one

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with co-finite in ind A, quasi-tilted algebras and generalized double tilted algebras, have been extensively investigated. The aim of the paper is to show that these two classes of algebras exhaust the class of all artin algebras A for which is co-finite in ind A, and derive some consequences. Dedicated to Stanislaw Balcerzyk on the occation of his 70th birthday     

Concealed-Canonical Algebras and Separating Tubular Families

We characterise those concealed-canonical algebras which ariseas endomorphism rings of tilting modules, all of whose indecomposablesummands have strictly positive rank, as those artin algebraswhose module categories have a separating exact subcategory(that is, a separating tubular family of standard tubes). This paper develops further the technique of shift automorphismswhich arises from the tubular structure. It is related to the characterisation of hereditary noetheriancategories with a tilting object as the categories of coherentsheaves on a weighted projective line. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.