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Flat and Ghost dimensions in tensor triangulated categories with a compact tilting generator

  • Xiaoyan Yang
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Abstract

Let 𝒯 be a tensor triangulated category with a compact tilting generator M and abelian heart β„‹M = MβŠ₯β‰ 0. We define a triangulated flat dimension flatdim𝒯X for Xβˆˆπ’―, and give criteria in terms of tensor vanishing and resolutions by the class of pure quotients of Add(M). We further introduce M-ghost length and M-ghost dimension, and show that the global dimension of β„‹M controls the M-ghost length of all objects in 𝒯, and that the M-ghost dimension of 𝒯 coincides with the weak global dimension of β„‹M and the supremum of flatdim𝒯X+infX of bounded objects, and the relative triangulated dimension of 𝒯c with respect to add(M). This extends the classical Hovey-Lockridge theorem, which is recovered for derived categories of rings.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026flat,
  title = {{Flat and Ghost dimensions in tensor triangulated categories with a compact tilting generator}},
  author = {Xiaoyan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14420},
  eprint = {2610.14420},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14420},
}
RIS
TY  - UNPB
TI  - Flat and Ghost dimensions in tensor triangulated categories with a compact tilting generator
AU  - Xiaoyan Yang
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14420
UR  - https://arxiv.org/abs/2610.14420
AB  - Let 𝒯 be a tensor triangulated category with a compact tilting generator M and abelian heart β„‹_M = M^βŠ₯_β‰ 0. We define a triangulated flat dimension flatdim_𝒯X for Xβˆˆπ’―, and give criteria in terms of tensor vanishing and resolutions by the class of pure quotients of Add(M). We further introduce M-ghost length and M-ghost dimension, and show that the global dimension of β„‹_M controls the M-ghost length of all objects in 𝒯, and that the M-ghost dimension of 𝒯 coincides with the weak global dimension of β„‹_M and the supremum of flatdim_𝒯X+infX of bounded objects, and the relative triangulated dimension of 𝒯^c with respect to add(M). This extends the classical Hovey-Lockridge theorem, which is recovered for derived categories of rings.
ER  -