Constructions of stable equivalences of Morita type for finite dimensional algebras II

2005 ◽  
Vol 251 (1) ◽  
pp. 21-39 ◽  
Author(s):  
Yuming Liu ◽  
Changchang Xi
Keyword(s):  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type ◽  
Finite Dimensional Algebras

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.

Rigidity dimension of algebras

2019 ◽  
pp. 1-27
Author(s):  
HONGXING CHEN ◽  
MING FANG ◽  
OTTO KERNER ◽  
STEFFEN KOENIG ◽  
KUNIO YAMAGATA
Keyword(s):  
Hochschild Cohomology ◽  
Global Dimension ◽  
Homological Dimension ◽  
Dominant Dimension ◽  
Finite Global Dimension ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Morita Type

Abstract A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. In particular, the rigidity dimension of a non-semisimple group algebra is finite and bounded by the order of the group. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.

Preservation of elementary equivalence under scalar extension

1982 ◽  
Vol 47 (4) ◽  
pp. 734-738
Author(s):  
Bruce I. Rose
Keyword(s):  
Identity Element ◽  
Positive Answer ◽  
General Question ◽  
Division Algebras ◽  
Extension Field ◽  
Elementary Equivalence ◽  
Applied Model ◽  
Finite Dimensional ◽  
Positive Statements ◽  
Finite Dimensional Algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

Levi and Malcev Theorems for Finite-Dimensional Algebras from the Variety Defined by the Identities x2 = J( x,y, zu) =0

2021 ◽  
Vol 28 (01) ◽  
pp. 87-90
Author(s):  
Óscar Guajardo Garza ◽  
Marina Rasskazova ◽  
Liudmila Sabinina
Keyword(s):  
Lie Algebras ◽  
Present Article ◽  
Variety Of Algebras ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

We study the variety of binary Lie algebras defined by the identities [Formula: see text], where [Formula: see text] denotes the Jacobian of [Formula: see text], [Formula: see text], [Formula: see text]. Building on previous work by Carrillo, Rasskazova, Sabinina and Grishkov, in the present article it is shown that the Levi and Malcev theorems hold for this variety of algebras.

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