Separated monic representations I: Gorenstein-projective modules

Luo, Xiu-Hua; Zhang, Pu

doi:10.1016/j.jalgebra.2017.01.038

Cited by 33 publications

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References 30 publications

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“…Let X i and (P i , Q i , α i , β i , g i ) be given as in (8) and (9). en, f i : X i ⟶ X i+1 is a Λ-map if and only if f i is given in (10).…”

Section: Complete Projective Resolutionsmentioning

confidence: 99%

“…Proof. First, assume that f i : X i ⟶ X i+1 is a Λ-map, where f i is given in (10). Since the diagram…”

Section: Complete Projective Resolutionsmentioning

confidence: 99%

“…After two decades, Enochs and Jenda [2] generalized it to an arbitrary ring and called it Gorenstein-projective modules. After that, this class of modules got special attention and has been studied by many authors (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15]). Given an algebra A, it is a fundamental problem to determine all the indecomposable finitely generated Gorenstein-projective A-modules, which is usually not easy.…”

Section: Introductionmentioning

confidence: 99%

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Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Asefa

2021

Journal of Mathematics

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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 .

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“…Let X i and (P i , Q i , α i , β i , g i ) be given as in (8) and (9). en, f i : X i ⟶ X i+1 is a Λ-map if and only if f i is given in (10).…”

Section: Complete Projective Resolutionsmentioning

confidence: 99%

“…Proof. First, assume that f i : X i ⟶ X i+1 is a Λ-map, where f i is given in (10). Since the diagram…”

Section: Complete Projective Resolutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Asefa

2021

Journal of Mathematics

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“…(2) Using (1) and the fact that smon(Q, I, A) is a resolving subcategory of Λ-mod (see [LZ,Thm. 3.1]), the assertion can be easily proved: the argument is as follows.…”

Section: 2mentioning

confidence: 99%

Separated monic representations II: Frobenius subcategories and RSS equivalences

Zhang

Xiong

2019

Trans. Amer. Math. Soc.

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This paper aims at looking for Frobenius subcategories, via the separated monomorphism category smon(Q, I, X ); and on the other hand, to establish an RSS equivalence from smon(Q, I, X ) to its dual sepi(Q, I, X ). For a bound quiver (Q, I) and an algebra A, where Q is acyclic and I is generated by monomial relations, let Λ = A ⊗ k kQ/I. For any additive subcategory X of A-mod, we construct smon(Q, I, X ) combinatorially. This construction describe Gorenstein-projective Λ-modules as GP(Λ) = smon(Q, I, GP(A)). It admits a homological interpretation, and enjoys a reciprocity smon(Q, I, ⊥ T ) = ⊥ (T ⊗ kQ/I) for a cotilting A-module T . As an application, smon(Q, I, X ) has Auslander-Reiten sequences if X is resolving and contravariantly finite with X = A-mod. In particular, smon(Q, I, A) has Auslander-Reiten sequences. It also admits a filtration interpretation as smon(Q, I, X ) = Fil(X ⊗ P(kQ/I)), provided that X is extension-closed. As an application, smon(Q, I, X )is an extension-closed Frobenius subcategory if and only if so is X . This gives "new" Frobenius subcategories of Λ-mod in the sense that they are not GP(Λ). Ringel-Schmidmeier-Simson equivalence smon(Q, I, X ) ∼ = sepi(Q, I, X ) is introduced and the existence is proved for arbitrary extension-closed subcategories X . In particular, the Nakayama functor N Λ gives an RSS equivalence smon(Q, I, A) ∼ = sepi(Q, I, A) if and only if A is Frobenius. For a chain Q with arbitrary I, an explicit formula of an RSS equivalence is found for arbitrary additive subcategories X .An advantage of taking B = kQ/I is that the representations of bound quiver (Q, I) ([R], [ARS], [ASS]) can be applied to study Λ-mod ([RS1-RS3], [S1-S3], [KLM1, KLM2], [RZ]). This choice of Λ = A ⊗ kQ/I is not restricted in the sense that, principally speaking, any algebra is of this form.A representation X of (Q, I) over A is a datum X = (X i , X α , i ∈ Q 0 , α ∈ Q 1 ), where X i ∈ Amod, and X α : X s(α) → X e(α) is an A-map, such that X γ := X α l · · · X α1 = 0 for each γ = α l · · · α 1 ∈ ρ, where ρ is a minimal set of generators of I. We call X i the i-th branch of X.

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“…Also Ringel and Schmidmeier have studied their Auslander-Reiten theory [22], as well as some particular cases of wild type [23]. Moreover they were studied with respect to Gorenstein-projective modules and tilting theory in [20,30]. The homological properties of submodule categories give extensive information on quiver Grassmannians, in particular their isomorphism classes correspond to strata in certain stratifications [6].…”

Section: Introductionmentioning

confidence: 99%

From submodule categories to the stable Auslander algebra

Eiriksson

2017

Journal of Algebra

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We construct two functors from the submodule category of a selfinjective representation-finite algebra Λ to the module category of the stable Auslander algebra of Λ. These functors factor through the module category of the Auslander algebra of Λ. Moreover they induce equivalences from the quotient categories of the submodule category modulo their respective kernels and said kernels have finitely many indecomposable objects up to isomorphism. Their construction uses a recollement of the module category of the Auslander algebra induced by an idempotent and this recollement determines a characteristic tilting and cotilting module. If Λ is taken to be a Nakayama algebra, then said tilting and cotilting module is a characteristic tilting module of a quasi-hereditary structure on the Auslander algebra. We prove that the self-injective Nakayama algebras are the only algebras with this property.

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Separated monic representations I: Gorenstein-projective modules

Cited by 33 publications

References 30 publications

Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Separated monic representations II: Frobenius subcategories and RSS equivalences

From submodule categories to the stable Auslander algebra

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