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The tensor triangular geometry of valuation domains

  • Tobias Barthel
  • Drew Heard
  • Beren Sanders
  • Changhan Zou
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Abstract

We study the tensor triangular geometry of derived categories of valuation domains, obtaining complete ring-theoretic characterizations of their main structural properties. In particular, although valuation domains are almost always non-noetherian, their derived categories always satisfy the local-to-global principle and hence the detection property. We establish precise criteria for homological detection, stratification, and homological stratification, revealing a clear hierarchy among these properties. Notably, we obtain examples of derived categories of commutative rings that are homologically stratified and satisfy the local-to-global principle, but are not stratified. We also determine when the category of modules over a valuation domain has Gabriel dimension, and use this to provide a negative answer to a question of Balchin, Gómez and Stevenson. Overall, valuation domains provide a rich class of non-noetherian rings whose derived categories can be understood in considerable depth.

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Cite

BibTeXRIS
BibTeX
@misc{barthel2026tensor,
  title = {{The tensor triangular geometry of valuation domains}},
  author = {Tobias Barthel and Drew Heard and Beren Sanders and Changhan Zou},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05298},
  eprint = {2611.05298},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05298},
}
RIS
TY  - UNPB
TI  - The tensor triangular geometry of valuation domains
AU  - Tobias Barthel
AU  - Drew Heard
AU  - Beren Sanders
AU  - Changhan Zou
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05298
UR  - https://arxiv.org/abs/2611.05298
AB  - We study the tensor triangular geometry of derived categories of valuation domains, obtaining complete ring-theoretic characterizations of their main structural properties. In particular, although valuation domains are almost always non-noetherian, their derived categories always satisfy the local-to-global principle and hence the detection property. We establish precise criteria for homological detection, stratification, and homological stratification, revealing a clear hierarchy among these properties. Notably, we obtain examples of derived categories of commutative rings that are homologically stratified and satisfy the local-to-global principle, but are not stratified. We also determine when the category of modules over a valuation domain has Gabriel dimension, and use this to provide a negative answer to a question of Balchin, Gómez and Stevenson. Overall, valuation domains provide a rich class of non-noetherian rings whose derived categories can be understood in considerable depth.
ER  -