Strongly Gorenstein-injective modules over Morita rings
Journal of Algebra and Its Applications, Jun 16, 2023
In this paper, we consider Morita rings with zero bimodule homomorphisms. We establish necessary ... more In this paper, we consider Morita rings with zero bimodule homomorphisms. We establish necessary and sufficient conditions for all strongly complete injective resolutions over a Morita ring [Formula: see text]. We also obtain necessary and sufficient conditions for all strongly Gorenstein-injective modules over [Formula: see text].
Let Δ (0,0) � A AN B BM A B be a Morita ring such that the bimodule homomorphisms are zero. In th... more Let Δ (0,0) � A AN B BM A B be a Morita ring such that the bimodule homomorphisms are zero. In this paper, we give sufficient conditions for a Δ (0,0)-module (X, Y, f, g) to be Gorenstein-projective. As an application, we give sufficient conditions when the algebras A and B inherit the strongly CM-freeness of Δ (0,0) .
Gorenstein-projective module is an important research topic in relative homological algebra, repr... more Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A, how to construct all the Gorenstein-projective A-modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ � A A M B 0 B. We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ � A A M B 0 B .
Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate ... more Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].
Construction of Gorenstein-projective modules over Morita rings
Journal of Algebra and Its Applications, Aug 30, 2022
In this paper, we obtain necessary and sufficient conditions for all complete projective resoluti... more In this paper, we obtain necessary and sufficient conditions for all complete projective resolutions over a Morita ring [Formula: see text]. As special cases, we get a class of complete projective resolutions over [Formula: see text], from the ones over [Formula: see text] and [Formula: see text]. We also obtain necessary and sufficient conditions for all Gorenstein-projective modules over a Morita ring [Formula: see text]. As special cases, we get a class of Gorenstein-projective modules over [Formula: see text], from the ones over [Formula: see text] and [Formula: see text]. Furthermore, we determine all complete projective resolutions as well as all Gorenstein-projective modules over the [Formula: see text] matrix algebra [Formula: see text] over [Formula: see text].
Let Δ 0,0 = A A N B B M A B be a Morita ring such that the bimodule homomorphisms are zero. In th... more Let Δ 0,0 = A A N B B M A B be a Morita ring such that the bimodule homomorphisms are zero. In this paper, we give sufficient conditions for a Δ 0,0 -module X , Y , f , g to be Gorenstein-projective. As an application, we give sufficient conditions when the algebras A and B inherit the strongly CM-freeness of Δ 0,0 .
Gorenstein-projective module is an important research topic in relative homological algebra, repr... more Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra A , how to construct all the Gorenstein-projective A -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ = A M B A 0 B . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ = A M B A 0 B .
Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate ... more Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].
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