The tensor triangular geometry of valuation domains

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.