Δ-Filtrations and Projective Resolutions for the Auslander–Dlab–Ringel Algebra
TL;DR: In this paper, the Δ-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of R USQ -modules are investigated.
Abstract: The ADR algebra R
A
of an Artin algebra A is a right ultra strongly quasihereditary algebra (RUSQ algebra). In this paper we study the Δ-filtrations of modules over RUSQ algebras and determine the projective covers of a certain class of R
A
-modules. As an application, we give a counterexample to a claim by Auslander–Platzeck–Todorov, concerning projective resolutions over the ADR algebra.
Citations
More filters
TL;DR: In this paper, it was shown that the Ringel dual of the ADR algebra R A is Morita equivalent to (R A o p ) o p if and only if all projective and injective A-modules are rigid and have the same Loewy length.
10 citations
TL;DR: In this article, a quasi-hereditary endomorphism was defined over a new class of finite dimensional monomial algebras with a special ideal structure, and the main result is a uniform formula describing the Ringel duals of these quasihereditary algesbras.
Abstract: We introduce quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-hereditary algebras.
As special cases, we obtain a Ringel-duality formula for a family of strongly quasi-hereditary algebras arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander-Dlab-Ringel algebras, and for Eiriksson and Sauter's nilpotent quiver algebras when the quiver has no sinks and no sources. We also recover Tan's result that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual.
3 citations
TL;DR: In this paper, the authors proved that the Auslander-Dlab-Ringel (ADR) algebras of semilocal modules are always left-strongly quasi-hereditary.
Abstract: Lin and Xi introduced Auslander–Dlab–Ringel (ADR) algebras of semilocal modules as a generalization of original ADR algebras and showed that they are quasi-hereditary. In this paper, we prove that such algebras are always left-strongly quasi-hereditary. As an application, we give a better upper bound for the global dimension of ADR algebras of semilocal modules. Moreover, we describe characterizations of original ADR algebras to be strongly quasi-hereditary.
1 citations
02 May 2018
TL;DR: In this paper, the authors studied quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algesbras with a special ideal structure.
Abstract: We study quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-hereditary algebras. As special cases, we obtain a Ringel duality formula for a family of strongly quasi-hereditary algebras arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander–Dlab–Ringel algebras, and for Eiriksson and Sauter’s nilpotent quiver algebras when the quiver has no sinks and no sources. We also recover Tan’s result that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual.
References
More filters
Book•
11 May 2010TL;DR: Artin rings as mentioned in this paper have been used to represent morphisms in the Auslander-Reiten-quiver and the dual transpose and almost split sequences, and they have been shown to be stable equivalence.
Abstract: 1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.
2,044 citations
TL;DR: The first of a series of papers dealing with the representation theory of artin algebras is presented in this paper, where the main purpose is to develop terminology and background material which will be used in the rest of the papers in the series.
Abstract: This is the first of a series of papers dealing with the representation theory of artin algebras, where by an artin algebra we mean an artin ring having the property that its center is an artin ring and λ is a finitely generated module over its center. The over all purpose of this paper is to develop terminology and background material which will be used in the rest of the papers in the series. While it is undoubtedly true that much of this material can be found in the literature or easily deduced from results already in the literature, the particular development presented here appears to be new and is especially well suited as a foundation for the papers to come.
1,267 citations
TL;DR: The Representation Theory of Artin Algebras II as mentioned in this paper has been applied to the representation theory of algebraic geometry, and it has been shown to be useful in algebraic representation theory.
Abstract: (1974). Representation Theory of Artin Algebras II. Communications in Algebra: Vol. 1, No. 4, pp. 269-310.
471 citations
423 citations
275 citations