Tate cohomology with respect to semidualizing modules

Sather-Wagstaff, Sean; Sharif, Tirdad; White, Diana

doi:10.1016/j.jalgebra.2010.07.007

Cited by 49 publications

(33 citation statements)

References 19 publications

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“…For more details about semidualizing modules and their related categories, see [2,5,11,13,16,19,23]. In [23], the authors gave some characterizations of some classical rings by studying the class of modules relative to semidualizing modules.…”

Section: Introductionmentioning

confidence: 99%

Cotorsion dimensions relative to semidualizing modules

Chen

2016

J. Algebra Appl.

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Communicated by J. TrlifajLet C be a semidualizing R-module, where R is a commutative ring. We first introduce the definition of C-cotorsion modules, and obtain the properties of C-cotorsion modules. As applications, we give some new characterizations for perfect rings. Second, we study the Foxby equivalences between the subclasses of the Auslander class and that of the Bass class with respect to C. Finally, we discuss C-cotorsion dimensions and investigate the transfer properties of strongly C-cotorsion dimensions under almost excellent extensions.J. Algebra Appl. Downloaded from www.worldscientific.com by STOCKHOLM UNIVERSITY on 08/27/15. For personal use only. X. Chen & J. ChenR (F, M ) = 0 for any flat R-module F . The class of cotorsion modules contains all pure-injective (hence injective) modules. An important feature of flat covers (respectively, cotorsion envelopes) is that their kernels (respectively, cokernels) are cotorsion (respectively, flat) by Wakamatsu's Lemmas [21, Sec. 2.1]). The existence of a flat cover and a cotorsion envelope for any module over any associative ring has been proved in [3]. On the other hand, cotorsion (and flat) modules have turned out to be very useful in characterizing rings. From [17], we know some properties about flat cotorsion modules with respect to semidualizing modules and applications in the properties of categories. Inspired by the above works, it is natural to ask what are the properties of cotorsion dimensions with respect to semidualizing modules, and what do they say about the characterizations of rings? The above problem is the main goal of this paper.In Sec. 2, we recall some basic concepts and properties about semidualizing modules and related modules. We introduce the definition of C-cotorsion modules over commutative rings, discuss the properties of C-cotorsion modules and establish the relation between C-cotorsion modules and cotorsion modules. It is proved that an R-module M is C-cotorsion if and only if M ∈ C ⊥1 and Hom R (C, M ) is a cotorsion R-module. Similarly, the relation between strongly C-cotorsion modules and cotorsion modules is given. As applications, we give some new equivalent characterizations of perfect rings. We prove that R is perfect if and only if C ⊥1 ⊆ Cot C (R). And we prove that the class of cotorsion R-modules in the Auslander class A C (R) and the class of C-cotorsion R-modules in the Bass class B C (R) are equivalent under Foxby equivalence. Finally, for a coherent ring R, it is shown that R is perfect if and only if every C-flat C-cotorsion R-module is C-projective if and only if every C-flat R-module is a pure submodule of some C-projective R-module.Section 3 is devoted to investigate the cotorsion dimensions with respect to semidualizing modules, i.e. strongly C-cotorsion dimensions. And we study transfer properties of the class of strongly C-cotorsion dimensions under almost excellent extensions. Definitions and General ResultsAt the beginning of this section, we recall some known notions and facts needed in the sequel.Let...

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Section: Introductionmentioning

confidence: 99%

Cotorsion dimensions relative to semidualizing modules

Chen

2016

J. Algebra Appl.

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“…Since then many authors have studied this theory in different abelian categories. For example, Veliche [15] and Christensen and Jørgensen [2] studied a Tate cohomology theory for complexes and Sather-Wagstaff et al [11] constructed a theory of Tate cohomology in any abelian category A based on a so-called Tate W-resolution, where W is a class of objects of A. The parallel theory of Tate homology has been treated by Iacob [8] and Christensen and Jørgensen [2,3]. Recently, Liang [10] develop a theory of Tate homology based on so-called Tate flat resolutions.…”

Section: Introductionmentioning

confidence: 99%

Gorenstein $$n$$ n -flat dimension and Tate homology

Selvaraj

Udhayakumar

2015

Boll Unione Mat Ital

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“…It is well known that the class of modules of Gorenstein dimension zero and that of Gorenstein projective modules coincide for finitely generated modules over a left and right Noetherian ring, and that Gorenstein projective modules and Gorenstein injective modules share many nice properties of projective modules and injective modules, respectively (cf. [10,11,14] Gorenstein projective and injective modules and some related generalized versions have been studied by many authors, see [1,4,[6][7][8][10][11][12][13][14][15][16][17][18][19][20][21], and the literatures listed in them.…”

Section: Introductionmentioning

confidence: 99%

Proper resolutions and Gorenstein categories

Huang

2013

Journal of Algebra

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Let A be an abelian category and C an additive full subcategory of A . We provide a method to construct a proper C -resolution (resp. coproper C -coresolution) of one term in a short exact sequence in A from that of the other two terms. By using these constructions, we answer affirmatively an open question on the stability of the Gorenstein category G(C ) posed by Sather-Wagstaff, Sharif and White; and also prove that G(C ) is closed under direct summands. In addition, we obtain some criteria for computing the C -dimension and the G(C )-dimension of an object in A .

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Tate cohomology with respect to semidualizing modules

Cited by 49 publications

References 19 publications

Cotorsion dimensions relative to semidualizing modules

Cotorsion dimensions relative to semidualizing modules

Gorenstein $$n$$ n -flat dimension and Tate homology

Proper resolutions and Gorenstein categories

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