Abstract

For a big tt-category, we give a characterization of the telescope conjecture $(TC)$ in terms of definable $\otimes$-ideals generated by homological residue fields. We formulate a stalk-locality property of $(TC)$ and prove that it holds in the case of the derived category of a quasicompact quasiseparated scheme, strengthening the 2024 result from Hrbek, Hu and Zhu. As an application, we find strong links between $(TC)$ and separation properties of the adic topology on local rings. This allows us to recover known examples and counterexamples of when $(TC)$ holds over a scheme, as well as to construct some new ones.

Keywords

Mathematical Subject Classification

Milestones

Authors