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Resolving dualities and applications to homological invariants

Part of: Rings and algebras with additional structure Homological algebra Homological methods Representation theory of rings and algebras

Published online by Cambridge University Press:  22 October 2024

Hongxing Chen  and
Jiangsheng Hu
Hongxing Chen
Affiliation:
School of Mathematical Sciences & Academy for Multidisciplinary Studies, Capital Normal University, Beijing 100048, P. R. China e-mail: [email protected]
Jiangsheng Hu*
Affiliation:
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, P. R. China
*
e-mail: [email protected]
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Abstract

Dualities of resolving subcategories of module categories over rings are introduced and characterized as dualities with respect to Wakamatsu tilting bimodules. By restriction of the dualities to smaller resolving subcategories, sufficient and necessary conditions for these bimodules to be tilting are provided. This leads to the Gorenstein version of both the Miyashita’s duality and Huisgen-Zimmermann’s correspondence. An application of resolving dualities is to show that higher algebraic K-groups and semi-derived Ringel–Hall algebras of finitely generated Gorenstein-projective modules over Artin algebras are preserved under tilting.

Keywords

DualityGorenstein-projective moduleresolving subcategorysemi-derived Ringel–Hall algebraWakamatsu tilting module

MSC classification

Primary: 16G10: Representations of Artinian rings 18G25: Relative homological algebra, projective classes 16W22: Actions of groups and semigroups; invariant theory
Secondary: 16E30: Homological functors on modules (Tor, Ext, etc.) 16E20: Grothendieck groups, $K$-theory, etc.
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 29
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This research was supported by the National Natural Science Foundation of China (Grant Nos. 12122112, 12031014, and 12171206). Also, the corresponding author J. S. Hu thanks Jiangsu 333 Project for partial support.

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