Localisations of rings of weak global dimension at most one

Abstract

We characterise the commutative rings of weak global dimension at most one whose derived categories can be stratified by the Balmer-Favi support theory. For such rings, this yields a bijective correspondence between the class of localising subcategories of the derived category, and the subsets of the Zariski spectrum. For a large class of rings of weak global dimension at most one including all valuation domains, this occurs if and only if the ring is locally strongly discrete, if and only if the Telescope conjecture holds, and if and only if the module category has Gabriel dimension. However, we also give an example of a valuation domain where all these conditions fail, and yet the residue fields generate its derived category and we can classify all the localising subcategories of $D(R)$. Additionally, in the same example, we show that the Balchin-Stevenson smashing support cannot stratify $D(R)$ either.