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Gorenstein projective objects in comma categories

Abstract

Let \(\mathcal {A}\) and \(\mathcal {B}\) be abelian categories and \({\mathbf {F}} :\mathcal {A}\rightarrow \mathcal {B}\) an additive and right exact functor which is perfect, and let \(({\mathbf {F}},\mathcal {B})\) be the left comma category. We give an equivalent characterization of Gorenstein projective objects in \(({\mathbf {F}},\mathcal {B})\) in terms of Gorenstein projective objects in \(\mathcal {B}\) and \(\mathcal {A}\). We prove that there exists a left recollement of the stable category of the subcategory of \(({\mathbf {F}},\mathcal {B})\) consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in \(\mathcal {B}\) and \(\mathcal {A}\). Moreover, this left recollement can be filled into a recollement when \(\mathcal {B}\) is Gorenstein and \({\mathbf {F}}\) preserves projectives.

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Acknowledgements

This research was partially supported by NSFC (Grant No. 11971225) and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The authors thank the referee for useful suggestions.

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Correspondence to Zhaoyong Huang.

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Peng, Y., Zhu, R. & Huang, Z. Gorenstein projective objects in comma categories. Period Math Hung 84, 186–202 (2022). https://doi.org/10.1007/s10998-021-00398-7

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  • DOI: https://doi.org/10.1007/s10998-021-00398-7

Keywords

  • Gorenstein objects
  • Comma categories
  • Perfect functors
  • Recollements

Mathematics Subject Classification

  • 18G25
  • 18E10
  • 18E30