scholarly journals On a Lift of an Individual Stable Equivalence to a Standard Derived Equivalence for Representation-Finite Self-injective Algebras

2003 ◽  
Vol 6 (4) ◽  
pp. 427-447 ◽  
Author(s):  
Hideto Asashiba
Keyword(s):  
Derived Equivalence ◽  
Stable Equivalence

scholarly journals Külshammer ideals of algebras of quaternion type

2018 ◽  
Vol 17 (08) ◽  
pp. 1850157
Author(s):  
Alexander Zimmermann
Keyword(s):  
Symmetric Algebra ◽  
Representation Type ◽  
Group Algebras ◽  
Closed Formula ◽  
Derived Equivalence ◽  
Joint Work ◽  
Stable Equivalence ◽  
Semidihedral Type ◽  
Quaternion Type ◽  
Morita Type

For a symmetric algebra [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text] Külshammer constructed a descending sequence of ideals of the center of [Formula: see text]. If [Formula: see text] is perfect, this sequence was shown to be an invariant under derived equivalence and for algebraically closed [Formula: see text] the dimensions of their image in the stable center were shown to be invariant under stable equivalence of Morita type. Erdmann classified algebras of tame representation type which may be blocks of group algebras, and Holm classified Erdmann’s list up to derived equivalence. In both classifications, certain parameters occur in the classification, and it was unclear if different parameters lead to different algebras. Erdmann’s algebras fall into three classes, namely of dihedral, semidihedral and of quaternion type. In previous joint work with Holm, we used Külshammer ideals to distinguish classes with respect to these parameters in case of algebras of dihedral and semidihedral type. In the present paper, we determine the Külshammer ideals for algebras of quaternion type and distinguish again algebras with respect to certain parameters.

scholarly journals Derived equivalences and stable equivalences of Morita type, I

2010 ◽  
Vol 200 ◽  
pp. 107-152 ◽  
Author(s):  
Wei Hu ◽  
Changchang Xi
Keyword(s):  
Derived Categories ◽  
Type I ◽  
Sufficient Condition ◽  
Derived Equivalence ◽  
Stable Equivalence ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type ◽  
Finite Dimensional Algebras

AbstractFor self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalenceFbetween the derived categories of Artin algebrasAandBarises naturally as a functorbetween their stable module categories, which can be used to compare certain homological dimensions ofAwith that ofB. We then give a sufficient condition for the functorto be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

scholarly journals Derived equivalences and stable equivalences of Morita type, I

2010 ◽  
Vol 200 ◽  
pp. 107-152 ◽  
Author(s):  
Wei Hu ◽  
Changchang Xi
Keyword(s):  
Derived Categories ◽  
Type I ◽  
Sufficient Condition ◽  
Derived Equivalence ◽  
Stable Equivalence ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type ◽  
Finite Dimensional Algebras

AbstractFor self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalence F between the derived categories of Artin algebras A and B arises naturally as a functor between their stable module categories, which can be used to compare certain homological dimensions of A with that of B. We then give a sufficient condition for the functor to be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

A note on the stable equivalence conjecture

1996 ◽  
Vol 24 (9) ◽  
pp. 2793-2809 ◽  
Author(s):  
Aiping Tang
Keyword(s):  
Stable Equivalence

On Derived Equivalence of Algebras of Semidihedral Type with Two Simple Modules

2018 ◽  
Vol 232 (5) ◽  
pp. 635-646
Author(s):  
A. I. Generalov ◽  
A. A. Zaikovskii
Keyword(s):  
Derived Equivalence ◽  
Simple Modules ◽  
Semidihedral Type

scholarly journals Classification of Symmetric Special Biserial Algebras With At Most One Non-Uniserial Indecomposable Projective

2015 ◽  
Vol 58 (3) ◽  
pp. 739-767 ◽  
Author(s):  
Nicole Snashall ◽  
Rachel Taillefer
Keyword(s):  
Natural Generalization ◽  
Stable Equivalence ◽  
Symmetric Algebras ◽  
Special Biserial Algebras ◽  
Weakly Symmetric ◽  
Dihedral Type ◽  
Indecomposable Projective Module ◽  
Algebras Of Dihedral Type ◽  
Euclidean Type

AbstractWe consider a natural generalization of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the weakly symmetric algebras of Euclidean type n, as studied by Bocian et al., as well as some algebras of dihedral type.

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