On semidualizing modules of ladder determinantal rings

We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate ℱC-resolutions, where C is a semidualizing module. We show firstly that the class of modules admitting a Tate ℱC-resolution is equal to the class of modules of finite 𝒢(ℱC)-projective dimension. Then an Avramov–Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather–Wagstaff et al. [Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010) 571–600], we establish a balance result for such Tate homology over a Cohen–Macaulay ring with a dualizing module by using a good conclusion provided in [E. E. Enochs, S. E. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. London Math. Soc. 44 (2012) 439–442].

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WEAK AB-CONTEXT FOR FP-INJECTIVE MODULES WITH RESPECT TO SEMIDUALIZING MODULES

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500394 ◽

2013 ◽

Vol 12 (07) ◽

pp. 1350039 ◽

Cited By ~ 2

Author(s):

JIANGSHENG HU ◽

DONGDONG ZHANG

Keyword(s):

Noetherian Ring ◽

Model Structure ◽

Injective Dimension ◽

Commutative Noetherian Ring ◽

New Model ◽

Injective Modules ◽

Semidualizing Modules ◽

Semidualizing Bimodule ◽

Gorenstein Injective

Let S and R be rings and SCR a semidualizing bimodule. We define and study GC-FP-injective R-modules, and these modules are exactly C-Gorenstein injective R-modules defined by Holm and Jørgensen provided that S = R is a commutative Noetherian ring. We mainly prove that the category of GC-FP-injective R-modules is a part of a weak AB-context, which is dual of weak AB-context in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander–Buchweitz approximations for R-modules with finite GC-FP-injective dimension. As an application, a new model structure in Mod R is given.

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BALANCE FOR TATE COHOMOLOGY WITH RESPECT TO SEMIDUALIZING MODULES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788713000232 ◽

2013 ◽

Vol 95 (2) ◽

pp. 223-240 ◽

Cited By ~ 5

Author(s):

LI LIANG ◽

GANG YANG

Keyword(s):

Commutative Ring ◽

Tate Cohomology ◽

Semidualizing Modules

AbstractIn this paper, we further study Tate cohomology of modules over a commutative ring with respect to semidualizing modules using the ideals of Sather-Wagstaff et al. [‘Tate cohomology with respect to semidualizing modules’, J. Algebra 324 (2010), 2336–2368]. In particular, we prove a balance result for the Tate cohomology ${\widehat{\mathrm{Ext} }}^{n} $ for any $n\in \mathbb{Z} $. This result complements the work of Sather-Wagstaff et al., who proved that the result holds for any $n\geq 1$. We also discuss some vanishing properties of Tate cohomology.

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AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

Algebras and Representation Theory ◽

10.1007/s10468-009-9195-9 ◽

2009 ◽

Vol 14 (3) ◽

pp. 403-428 ◽

Cited By ~ 37

Author(s):

Sean Sather-Wagstaff ◽

Tirdad Sharif ◽

Diana White

Keyword(s):

Flat Modules ◽

Semidualizing Modules ◽

Gorenstein Flat

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Presentations of rings with a chain of semidualizing modules

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-96668 ◽

2017 ◽

Vol 121 (2) ◽

pp. 161

Author(s):

Ensiyeh Amanzadeh ◽

Mohammad T. Dibaei

Keyword(s):

Local Ring ◽

Gorenstein Ring ◽

Semidualizing Modules ◽

Dualizing Module ◽

A Chain

Inspired by Jorgensen et al., it is proved that if a Cohen-Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$-modules of length $n$, then $R\cong Q/(I_1+\cdots +I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\dots , I_n$ of $Q$; and, for each $\Lambda \subseteq [n]$, the ring $Q/(\sum _{\ell \in \Lambda } I_\ell )$ has some interesting cohomological properties. This extends the result of Jorgensen et al., and also of Foxby and Reiten.

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