Recollement ideals and recollements of Gorenstein defect categories

Huanhuan Li; Hanyang You; Yuefei Zheng; Haiyan Zhu; Huanhuan Li; Hanyang You; Yuefei Zheng; Haiyan Zhu

doi:10.3934/era.2026106

Electronic Research Archive

2026, Volume 34, Issue 4: 2348-2363. doi: 10.3934/era.2026106

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Research article

Recollement ideals and recollements of Gorenstein defect categories

1.
School of Mathematics and Statistics, Xidian University, Xi'an 710071, China
2.
College of Science, Shanghai University, Shanghai 200444, China
3.
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
4.
College of Science, Northwest A&F University, Yangling 712100, China
5.
College of Science, Zhejiang University of Technology, Hangzhou 310000, China

Received: 03 December 2025 Revised: 27 February 2026 Accepted: 03 March 2026 Published: 17 March 2026

Let $ R $ be an Artin algebra and $ e\in R $ an idempotent. We study when the recollement ideal $ ReR $ induces (left) recollement of Gorenstein defect categories. As a consequence of this phenomenon, concrete triangle-equivalences between Gorenstein defect categories (resp. stable categories of Gorenstein projective modules) of $ R $ and $ R/ReR $ (resp. $ eRe $) are obtained, and reduction conditions for Gorensteinness (resp. CM-freeness) of $ R $ and Gorenstein projective conjecture over $ R $ are provided. Finally, some applications in the context of triangular matrix algebras are given.
- recollements,
- recollement ideals,
- Gorenstein projective modules,
- Gorenstein defect categories,
- Gorenstein projective conjecture
Citation: Huanhuan Li, Hanyang You, Yuefei Zheng, Haiyan Zhu. Recollement ideals and recollements of Gorenstein defect categories[J]. Electronic Research Archive, 2026, 34(4): 2348-2363. doi: 10.3934/era.2026106

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Abstract

Let $ R $ be an Artin algebra and $ e\in R $ an idempotent. We study when the recollement ideal $ ReR $ induces (left) recollement of Gorenstein defect categories. As a consequence of this phenomenon, concrete triangle-equivalences between Gorenstein defect categories (resp. stable categories of Gorenstein projective modules) of $ R $ and $ R/ReR $ (resp. $ eRe $) are obtained, and reduction conditions for Gorensteinness (resp. CM-freeness) of $ R $ and Gorenstein projective conjecture over $ R $ are provided. Finally, some applications in the context of triangular matrix algebras are given.

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