Definable coaisles over rings of weak global dimension at most one

TL;DR

This paper studies t-structures with definable coaisles in the derived category of rings with weak global dimension at most one, generalizing previous stable cases and providing classifications and conjecture analyses.

Contribution

It extends the classification of definable coaisles to non-stable t-structures over rings of weak global dimension at most one, especially in the commutative case.

Findings

Stable t-structures correspond to smashing localizations.

Definable coaisles over valuation domains are fully classified.

Not all definable coaisles come from homological ring epimorphisms.

Abstract

In the setting of the unbounded derived category D(R) of a ring R of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [B\v{S}17] to the non-stable case. As in the stable case [B\v{S}17], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the telescope conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied.