Auslander–Reiten conjecture and Auslander–Reiten duality

Celikbas, Olgur; Takahashi, Ryo

doi:10.1016/j.jalgebra.2013.02.007

Cited by 40 publications

(21 citation statements)

References 32 publications

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“…If Ext i R (M, M ⊕ R) = 0 for all i > 0, then M is projective. There are already some results in the study of classes of commutative rings satisfying (ARC); see for instance [3,9,15,16,21]. In the next result, by using the above corollary, we investigate ascent and descent of (ARC) between a local ring and its completion.…”

Section: Extension-closed Subcategories and Annihilation Of Cohomologymentioning

confidence: 85%

Annihilation of Cohomology, Generation of Modules and Finiteness of Derived Dimension

et al. 2016

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Abstract. Let (R, m, k) be a commutative noetherian local ring of Krull dimension d. We prove that the cohomology annihilator ca(R) of R is m-primary if and only if for some n ≥ 0 the n-th syzygies in mod R are constructed from syzygies of k by taking direct sums/summands and a fixed number of extensions. These conditions yield that R is an isolated singularity such that the bounded derived category D b (R) and the singularity category D sg (R) have finite dimension, and the converse holds when R is Gorenstein. We also show that the modules locally free on the punctured spectrum are constructed from syzygies of finite length modules by taking direct sums/summands and d extensions. This result is exploited to investigate several ascent and descent problems between R and its completion R.

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Section: Extension-closed Subcategories and Annihilation Of Cohomologymentioning

confidence: 85%

Annihilation of Cohomology, Generation of Modules and Finiteness of Derived Dimension

et al. 2016

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“…Motivated by this conjecture and its connection to the celebrated Auslander-Reiten Conjecture [7], Celikbas and Takahashi [20] introduced the following condition on a local ring (R, m):…”

Section: Thetas and Etas And Tors! Oh My!mentioning

confidence: 99%

Vanishing of Tor, and why we care about it

Celikbas¹,

Wiegand²

2015

Journal of Pure and Applied Algebra

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Abstract. Given finitely generated modules M and N over a local ring R, the tensor product M ⊗ R N typically has nonzero torsion. Indeed, the assumption that the tensor product is torsion-free influences the structure and vanishing of the modules Tor R i (M, N ) for all i ≥ 1. In turn, the vanishing of Tor R i (M, N ) imposes restrictions on the depth properties of the modules M and N . These connections made their first appearance in Auslander's 1961 paper "Modules over unramified regular local rings". We will survey the literature on these topics, with emphasis on progress during the past twenty years. This paper is dedicated to Hans-Bjorn Foxby in recognition of his huge impact on homological algebra, module theory, and commutative algebra.

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“…We refer to [AR,BFS,CT,FZ,LH,LJ,W1,W2,W3,X1,X2,X3,Y,Z] for details. The notion of excellent extension of rings was introduced by Passman in [P] which is important in studying the algebraic structure of group rings.…”

Section: Introductionmentioning

confidence: 99%

Real-time optical OFDM transmissions with spectral efficiency up to 6.93 bit/s/Hz over 50 km SSMF IMDD systems

Zhang

Liu

Chen

et al. 2017

Optics Communications

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Abstract. In this paper, we introduce the notion of excellent extension of rings. Let Γ be an excellent extension of an artin algebra Λ, we prove that Λ satisfies the Gorenstein symmetry conjecture (resp. finitistic dimension conjecture, AuslanderGorenstein conjecture, Nakayama conjecture) if and only if so does Γ. As a special case of excellent extensions, if G is a finite group whose order is invertible in Λ acting on Λ and Λ is G-stable, we prove that if the skew group algebras ΛG satisfies strong Nakayama conjecture (resp. generalized Nakayama conjecture), then so does Λ.

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Auslander–Reiten conjecture and Auslander–Reiten duality

Cited by 40 publications

References 32 publications

Annihilation of Cohomology, Generation of Modules and Finiteness of Derived Dimension

Annihilation of Cohomology, Generation of Modules and Finiteness of Derived Dimension

Vanishing of Tor, and why we care about it

Real-time optical OFDM transmissions with spectral efficiency up to 6.93 bit/s/Hz over 50 km SSMF IMDD systems

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