The Gorenstein-projective modules over a monomial algebra

Chen, Xiaowu; Shen, Dawei; Zhou, Guodong

doi:10.1017/s0308210518000185

Cited by 40 publications

(48 citation statements)

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“…Let Λ = kQ/I be a monomial algebra. Following [14], we say that a pair (p, q) of non-zero paths in Λ is a perfect pair provided that the following conditions are satisfied:…”

Section: Examplesmentioning

confidence: 99%

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Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

Bekkert

Giraldo

Vélez-Marulanda

2020

Journal of Pure and Applied Algebra

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Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Λ, V ). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Λ is self-injective that R(Λ, V ) is universal. In this paper, we prove instead that if Λ is arbitrary but V is Gorenstein-projective then R(Λ, V ) is also universal when the stable endomorphism ring of V is isomorphic to k. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if Λ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective Λ-module has a universal deformation ring that is isomorphic to either k or to k[[t]]/(t 2 ).2010 Mathematics Subject Classification. 16G10 and 16G20 and 20C20.

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“…Let Λ = kQ/I be a monomial algebra. Following [14], we say that a pair (p, q) of non-zero paths in Λ is a perfect pair provided that the following conditions are satisfied:…”

Section: Examplesmentioning

confidence: 99%

“…Assume that there is no overlap in Λ and let V be a finitely generated indecomposable Gorenstein-projective left Λ-module. If V is non-projective, then by the arguments in the proof of [14,Prop. 5.9] together with Lemma 2.4 we obtain that End Λ (V ) = k and Proof.…”

Section: Examplesmentioning

confidence: 99%

Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

Bekkert

Giraldo

Vélez-Marulanda

2020

Journal of Pure and Applied Algebra

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“…In the following example, we verify Theorem 1.2 with a particular basic connected Nakayama k-algebra with no simple projective modules in which there is no overlap (in the sense of [10]).…”

Section: Proof Of the Main Resultsmentioning

confidence: 63%

“…We refer the reader to look at e.g. [10,17] (and their references) for basic concepts concerning Gorenstein-projective modules. For basic concepts from the representation theory of algebras such as projective covers, syzygies of modules, stable categories and homological dimension of modules over finite dimensional algebras, we refer the reader to [2,3,12,25].…”

Section: Introductionmentioning

confidence: 99%

A note on deformations of Gorenstein-projective modules over finite dimensional algebras

Vélez-Marulanda

2019

rev.colomb.mat

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Let k be a fixed field of arbitrary characteristic, and let Λ be a finite dimensional k-algebra. Assume that V is a left Λ-module of finite dimension over k. F. M. Bleher and the author previously proved that V has a well-defined versal deformation ring R(Λ, V ) which is a local complete commutative Noetherian ring with residue field isomorphic to k. Moreover, R(Λ, V ) is universal if the endomorphism ring of V is isomorphic to k. In this article we prove that if Λ is a basic connected Nakayama algebra without simple modules and V is a Gorenstein-projective left Λ-module, then R(Λ, V ) is universal. Moreover, we also prove that the universal deformation rings R(Λ, V ) and R(Λ, ΩV ) are isomorphic, where ΩV denotes the first syzygy of V . This result extends the one obtained by F. M. Bleher and D. J. Wackwitz concerning universal deformation rings of finitely generated modules over self-injective Nakayama algebras. In addition, we also prove the following result concerning versal deformation rings of finitely generated modules over triangular matrix finite dimensional algebras. Let Σ = Λ B 0 Γ be a triangular matrix finite dimensional Gorenstein k-algebra with Γ of finite global dimension and B projective as a left Λ-module. If V W f is a finitely generated Gorenstein-projective left Σ-module, then the versal deformation rings R Σ, V W f and R(Λ, V ) are isomorphic.2010 Mathematics Subject Classification. 16G10 and 16G20 and 20C20. Key words and phrases. (Uni)versal deformation rings and finitely generated Gorenstein-projective modules and Nakayama algebras and triangular matrix algebras.

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“…Thus, one of the most important tasks is to describe Gorenstein projective modules, especially in non-commutative case. Up to now, there are some partial results, mainly concentrated on Artin algebras, such as T 2 -extension of an Artin algebra [LZ10], upper triangular matrix Artin algebra [XZ12,Zha13,EHSL16], Artin algebras with radical square zero [Che12a,RX12], Nakayama algebras [Rin13], monomial algebras [CSZ15], and some cases of tensor product of two algebras-we will show it explicitly below. In this paper, we study Gorenstein projective modules over the tensor product of two algebras.…”

Section: Introductionmentioning

confidence: 99%

Gorenstein projective bimodules via monomorphism categories and filtration categories

Luo

Xiong

et al. 2019

Journal of Pure and Applied Algebra

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We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras A and B, we use the special monomorphism category Mon(B, A-Gproj) to describe some Gorenstein projective bimodules over the tensor product of A and B. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for Mon(B, A-Gproj) being the category of all Gorenstein projective bimodules. In addition, if both A and B are Gorenstein, we can describe the category of all Gorenstein projective bimodules via filtration categories. Similarly, in this case, we get the same result for infinitely generated Gorenstein projective bimodules.

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The Gorenstein-projective modules over a monomial algebra

Abstract: We classify indecomposable non-projective Gorenstein-projective modules over a monomial algebra via the notion of perfect paths. We apply this classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

Cited by 40 publications

References 16 publications

Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras

A note on deformations of Gorenstein-projective modules over finite dimensional algebras

Gorenstein projective bimodules via monomorphism categories and filtration categories

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