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Approximability of Triangulated Categories under Separable Extensions

  • Mengzhen Zhu
  • Yuefei Zheng
  • Xianhui Fu
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Abstract

In this paper, we investigate the behavior of approximability under separable extensions of triangulated categories. Let ℬ be an approximable triangulated category and let T=(T,θ,μ) be a smashing, stably separable exact monad on ℬ. Assume that the category of T-modules T−Modℬ admits a canonical triangulated structure. We prove that if T is right t-exact with respect to a t-structure in the preferred equivalence class of ℬ, then T−Modℬ is again an approximable triangulated category. As an application, we show that, under suitable assumptions, a smashing localization functor on an approximable triangulated category induces a recollement of approximable triangulated categories, thereby yielding a converse-type result to the gluing theorem of Burke--Neeman--Pauwels. Finally, under suitable hypotheses, we prove that separable extensions restrict naturally to the intrinsic subcategories associated with approximable triangulated categories.

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Cite

BibTeXRIS
BibTeX
@misc{zhu2026approximability,
  title = {{Approximability of Triangulated Categories under Separable Extensions}},
  author = {Mengzhen Zhu and Yuefei Zheng and Xianhui Fu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27031},
  eprint = {2610.27031},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27031},
}
RIS
TY  - UNPB
TI  - Approximability of Triangulated Categories under Separable Extensions
AU  - Mengzhen Zhu
AU  - Yuefei Zheng
AU  - Xianhui Fu
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.27031
UR  - https://arxiv.org/abs/2610.27031
AB  - In this paper, we investigate the behavior of approximability under separable extensions of triangulated categories. Let ℬ be an approximable triangulated category and let T=(T,θ,μ) be a smashing, stably separable exact monad on ℬ. Assume that the category of T-modules T−Mod_ℬ admits a canonical triangulated structure. We prove that if T is right t-exact with respect to a t-structure in the preferred equivalence class of ℬ, then T−Mod_ℬ is again an approximable triangulated category. As an application, we show that, under suitable assumptions, a smashing localization functor on an approximable triangulated category induces a recollement of approximable triangulated categories, thereby yielding a converse-type result to the gluing theorem of Burke--Neeman--Pauwels. Finally, under suitable hypotheses, we prove that separable extensions restrict naturally to the intrinsic subcategories associated with approximable triangulated categories.
ER  -