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Duality pairs induced by Auslander and Bass classes

Duality pairs induced by Auslander and Bass classes AbstractLet R and S be arbitrary rings and let CSR{{}_{R}C_{S}}be a semidualizing bimodule, and let 𝒜C⁢(Rop){\mathcal{A}_{C}(R^{\mathrm{op}})}and ℬC⁢(R){\mathcal{B}_{C}(R)}be the Auslander and Bass classes, respectively. Then both pairs(𝒜C⁢(Rop),ℬC⁢(R)) and (ℬC⁢(R),𝒜C⁢(Rop))(\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(%\mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}}))are coproduct-closed and product-closed duality pairs and both𝒜C⁢(Rop){\mathcal{A}_{C}(R^{\mathrm{op}})}and ℬC⁢(R){\mathcal{B}_{C}(R)}are covering and preenveloping;in particular, the former duality pair is perfect. Moreover,if ℬC⁢(R){\mathcal{B}_{C}(R)}is enveloping in Mod⁡R{\operatorname{Mod}R}, then 𝒜C⁢(S){\mathcal{A}_{C}(S)}is enveloping in Mod⁡S{\operatorname{Mod}S}.Also, some applications to the Auslander projective dimension of modules are given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

Duality pairs induced by Auslander and Bass classes

Georgian Mathematical Journal , Volume 28 (6): 16 – Dec 1, 2021

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Publisher
de Gruyter
Copyright
© 2021 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1572-9176
eISSN
1572-9176
DOI
10.1515/gmj-2021-2101
Publisher site
See Article on Publisher Site

Abstract

AbstractLet R and S be arbitrary rings and let CSR{{}_{R}C_{S}}be a semidualizing bimodule, and let 𝒜C⁢(Rop){\mathcal{A}_{C}(R^{\mathrm{op}})}and ℬC⁢(R){\mathcal{B}_{C}(R)}be the Auslander and Bass classes, respectively. Then both pairs(𝒜C⁢(Rop),ℬC⁢(R)) and (ℬC⁢(R),𝒜C⁢(Rop))(\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(%\mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}}))are coproduct-closed and product-closed duality pairs and both𝒜C⁢(Rop){\mathcal{A}_{C}(R^{\mathrm{op}})}and ℬC⁢(R){\mathcal{B}_{C}(R)}are covering and preenveloping;in particular, the former duality pair is perfect. Moreover,if ℬC⁢(R){\mathcal{B}_{C}(R)}is enveloping in Mod⁡R{\operatorname{Mod}R}, then 𝒜C⁢(S){\mathcal{A}_{C}(S)}is enveloping in Mod⁡S{\operatorname{Mod}S}.Also, some applications to the Auslander projective dimension of modules are given.

Journal

Georgian Mathematical Journalde Gruyter

Published: Dec 1, 2021

Keywords: Duality pairs; Auslander classes; Bass classes; semidualizing bimodules; (pre)covers; (pre)envelopes; cotorsion pairs; Auslander projective dimension; 18G25; 16E10; 16E30

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