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On triangle equivalences of stable categories

Part of: Homological algebra Abelian categories Commutative algebra: Homological methods Homological methods

Published online by Cambridge University Press:  26 January 2019

Zhenxing Di ,
Zhongkui Liu  and
Jiaqun Wei
Zhenxing Di
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou730070, People's Republic of China (dizhenxing19841111@126.com; liuzk@nwnu.edu.cn)
Zhongkui Liu
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou730070, People's Republic of China (dizhenxing19841111@126.com; liuzk@nwnu.edu.cn)
Jiaqun Wei*
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou730070, People's Republic of China and Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing210023, People's Republic of China (weijiaqun@njnu.edu.cn)
*
*Corresponding author.
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Abstract

We apply the Auslander–Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga–Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga–Gorenstein rings and Gorenstein algebras.

Keywords

Presilting subcategorysubfactor triangulated categoryprojective Frobenius subcategoryGorenstein projective module

MSC classification

Primary: 13D05: Homological dimension
Secondary: 16E35: Derived categories 18G25: Relative homological algebra, projective classes 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 955 - 974
Copyright
Copyright © Royal Society of Edinburgh 2019

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Footnotes

In Memory of Professor Ragnar-Olaf Buchweitz

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