Grade, dominant dimension and Gorenstein algebras

Gao, Nan; Koenig, Steffen

doi:10.1016/j.jalgebra.2014.11.028

Cited by 8 publications

(5 citation statements)

References 18 publications

(20 reference statements)

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“…2 Nan Gao, Jing Ma and Juxia Zhang Gao and König [6] introduced gendo-d-Gorenstein algebras, which are as correspondents of Gorenstein algebras under a Morita-Tachikawa correspondence. It is known that the double centraliser property plays a key role in defining gendo-d-Gorenstein algebras.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Canonical Comultiplication and Double Centraliser Property

Gao

Zhang

2023

Taiwanese J. Math.

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In this paper, we show the existence of the attached comultiplication structure on Hom eAe (eA, D(Ae)) if an (eAe, A)-bimodule eA has the double centraliser property over an algebra A with the idempotent e. Then we apply it on gendo-Gorenstein algebras. As applications, we give a sufficient and necessary condition for a gendo-Gorenstein algebra to be Gorenstein, and give a bocs-theoretic characterisation of the double centraliser property.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Canonical Comultiplication and Double Centraliser Property

Gao

Zhang

2023

Taiwanese J. Math.

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“…The Morita-Tachikawa correspondence also brought interest to define and study many new classes of algebras for example: Morita algebras [23], gendo-Frobenius algebras [30] and gendo-symmetric algebras [14] as the counterparts through Morita-Tachikawa correspondence of self-injective, Frobenius and symmetric algebras, respectively. See also [15] for the counterparts of Gorenstein algebras under the Morita-Tachikawa correspondence. Recently, these correspondences started to make appearances also in the context of exact categories [13,16,18].…”

Section: Introductionmentioning

confidence: 99%

Higher Morita–Tachikawa correspondence

Cruz

2024

Bulletin of London Math Soc

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Important correspondences in representation theory can be regarded as restrictions of the Morita–Tachikawa correspondence. Moreover, this correspondence motivates the study of many classes of algebras like Morita algebras and gendo‐symmetric algebras. Explicitly, the Morita–Tachikawa correspondence describes that endomorphism algebras of generators–cogenerators over finite‐dimensional algebras are exactly the finite‐dimensional algebras with dominant dimension at least two. In this paper, we introduce the concepts of quasi‐generators and quasi‐cogenerators that generalise generators and cogenerators, respectively. Using these new concepts, we present higher versions of the Morita–Tachikawa correspondence that take into account relative dominant dimension with respect to a self‐orthogonal module with arbitrary projective and injective dimensions. These new versions also hold over Noetherian algebras that are finitely generated and projective over a commutative Noetherian ring.

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“…with objects considered up to isomorphism on each side [21,26]. This result is sometimes known [13,25] as the Morita-Tachikawa correspondence. The following definition will be convenient throughout the paper.…”

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confidence: 97%

“…By the Morita-Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.with objects considered up to isomorphism on each side [21,26]. This result is sometimes known [13,25] as the Morita-Tachikawa correspondence. The following definition will be convenient throughout the paper.Definition 1.1.…”

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confidence: 99%

Special tilting modules for algebras with positive dominant dimension

Pressland,

Sauter

2017

Preprint

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We study a set of uniquely determined tilting and cotilting modules for an algebra with positive dominant dimension, with the property that they are generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example that their endomorphism algebras always have global dimension at most that of the original algebra. We characterise d-Auslander-Gorenstein algebras and d-Auslander algebras via the property that the relevant tilting and cotilting modules coincide. By the Morita-Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.with objects considered up to isomorphism on each side [21,26]. This result is sometimes known [13,25] as the Morita-Tachikawa correspondence. The following definition will be convenient throughout the paper.Definition 1.1. A Morita-Tachikawa triple (A, E, Γ) consists of a finite-dimensional algebra A, a generating-cogenerating A-module E, and Γ ∼ = End A (E) op . Thus, assuming as we usually will that all objects are basic, the set of Morita-Tachikawa triples is the graph of the Morita-Tachikawa correspondence. Given a basic algebra Γ of dominant dimension at least 2, it appears in the (unique up to isomorphism) Morita-Tachikawa triple

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Grade, dominant dimension and Gorenstein algebras

Cited by 8 publications

References 18 publications

Canonical Comultiplication and Double Centraliser Property

Canonical Comultiplication and Double Centraliser Property

Higher Morita–Tachikawa correspondence

Special tilting modules for algebras with positive dominant dimension

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