The tensor triangular geometry of valuation domains

Tobias Barthel; Drew Heard; Beren Sanders; Changhan Zou

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arXiv · 2611.05298

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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Tobias Barthel, Drew Heard, Beren Sanders, Changhan Zou. 2026-11-03. The tensor triangular geometry of valuation domains. https://arxiv.org/abs/2611.05298

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