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

@article{Asashiba2003OnAL,
  title={On a Lift of an Individual Stable Equivalence to a Standard Derived Equivalence for Representation-Finite Self-injective Algebras},
  author={Hideto Asashiba},
  journal={Algebras and Representation Theory},
  year={2003},
  volume={6},
  pages={427-447}
}
  • H. Asashiba
  • Published 1 October 2003
  • Mathematics
  • Algebras and Representation Theory
We shall show that every stable equivalence (functor) between representation-finite self-injective algebras not of type (D3m,s/3,1) with m≥2, 3∤s lifts to a standard derived equivalence. This implies that all stable equivalences between these algebras are of Morita type. 
The liftability question for stable equivalences between representation-finite self-injective algebras
Let k be an algebraically closed field. It is known that any stable equivalence between standard representation-finite self-injective k-algebras (without block of Loewy length 2) lifts to a standard
Constructions of stable equivalences of Morita type for finite dimensional algebras II
Suppose k is a field. Let A and B be two finite dimensional k-algebras such that there is a stable equivalence of Morita type between A and B. In this paper, we prove that (1) if A and B are
Constructions of derived equivalences for algebras and rings
In this article, we shall survey some aspects of our recent (or related) constructions of derived equivalences for algebras and rings.
Derived equivalences and stable equivalences of Morita type, I
Abstract For 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
Derived equivalences of algebras
  • Changchang Xi
  • Mathematics
    Bulletin of the London Mathematical Society
  • 2018
Derived categories and equivalences between them are the pièce de résistance of modern homological algebra. They are widely used in many branches of mathematics, especially in algebraic geometry and
On Simple-Minded Systems Over Representation-Finite Self-Injective Algebras
Let $A$ be a representation-finite self-injective algebra over an algebraically closed field $k$. We give a new characterization for an orthogonal system in the stable module category $A$-$\stmod$ to
Auto-equivalences of stable module categories
We construct nontrivial auto-equivalences of stable module categories for elementary, local symmetric algebras over a field k. These auto-equivalences are modeled after the spherical twists of Seidel
Derived equivalences and stable equivalences of Morita type, II
We consider the question of lifting stable equivalences of Morita type to derived equivalences. One motivation comes from an approach to Broue’s abelian defect group conjecture. Another motivation is
Realizing orbit categories as stable module categories: a complete classification
We classify all triangulated orbit categories of path-algebras of Dynkin diagrams that are triangle equivalent to a stable module category of a representation-finite self-injective standard algebra.
...
1
2
3
4
...

References

SHOWING 1-10 OF 11 REFERENCES
Equivalences of Blocks of Group Algebras
Let O be a complete local noetherian ring, whose field of fractions has characteristic zero and residue field has non-zero characteristic. A block algebra over O is an indecomposable summand of the
Selfinjective and simply connected algebras
In this paper, we present a new approach to the problem of classifying all basic finite-dimensional algebras over an algebraically closed field k which are connected, selfinjective and
Derived Equivalences As Derived Functors
The Derived Equivalence Classification of Representation-Finite Selfinjective Algebras
Sous les catégories dérivées , C
  • Equivalences of blocks of group algebras , Finite dimensional algebras and related topics
  • 1994
On a theorem of E. Green on the dual of the transpose, Representations of finite dimensional algebras (Tsukuba
  • CMS Conf. Proc.,
  • 1990
Morita Theory for Derived Categories
...
1
2
...