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A subcategory with remarkable homological properties via actions of tensor categories

  • Kevin Coulembier
  • Edmund Heng
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Abstract

By equipping the category β„³ of modules over a finite dimensional algebra with the action of a finite tensor category π’ž, we introduce a subcategory 𝒫ℳ satisfying numerous remarkable properties: it is generating and cogenerating, functorially finite, closed under Auslander--Reiten translations (i.e. precluster tilting), and it computes the finitistic dimension. Through a functor category perspective, 𝒫ℳ is identified with the π’ž-module functors that are projective, and we also prove a π’ž-equivariant version of the Auslander(--Solberg) correspondence. As examples, we show that 𝒫ℳ contains only finitely many indecomposable modules for some tensor product algebras. For some family of 1-Iwanaga--Gorenstein algebras introduced by Geiss--Leclerc--SchrΓΆer, we show that 𝒫ℳ is exactly the subcategory of Ο„-locally free modules.

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Cite

BibTeXRIS
BibTeX
@misc{coulembier2026subcategory,
  title = {{A subcategory with remarkable homological properties via actions of tensor categories}},
  author = {Kevin Coulembier and Edmund Heng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17646},
  eprint = {2610.17646},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17646},
}
RIS
TY  - UNPB
TI  - A subcategory with remarkable homological properties via actions of tensor categories
AU  - Kevin Coulembier
AU  - Edmund Heng
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17646
UR  - https://arxiv.org/abs/2610.17646
AB  - By equipping the category β„³ of modules over a finite dimensional algebra with the action of a finite tensor category π’ž, we introduce a subcategory 𝒫ℳ satisfying numerous remarkable properties: it is generating and cogenerating, functorially finite, closed under Auslander--Reiten translations (i.e. precluster tilting), and it computes the finitistic dimension. Through a functor category perspective, 𝒫ℳ is identified with the π’ž-module functors that are projective, and we also prove a π’ž-equivariant version of the Auslander(--Solberg) correspondence. As examples, we show that 𝒫ℳ contains only finitely many indecomposable modules for some tensor product algebras. For some family of 1-Iwanaga--Gorenstein algebras introduced by Geiss--Leclerc--SchrΓΆer, we show that 𝒫ℳ is exactly the subcategory of Ο„-locally free modules.
ER  -