Telescope conjecture via homological residue fields with applications to schemes
Michal Hrbek
TL;DR
This work develops a comprehensive framework for the Telescope Conjecture $\mathsf{(TC)}$ in big tensor-triangulated categories by leveraging definable $\otimes$-ideals and Balmer–Krause–Stevenson’s homological residue fields. It introduces a stalk-locality principle that reduces global TC to data on stalks and, under NoSC, characterizes TC in terms of whether the homological residue fields generate the stalk categories as definable $\otimes$-ideals. The paper then applies these ideas to $\mathbf{D}(X)$, showing TC is a stalk-local property for quasi-compact, quasi-separated schemes and connecting residue-field data to $\mathfrak{m}$-adic separation properties, yielding both known and new examples. It also relates TC to ring epimorphisms and provides a restricted framework RTC for commutative rings, with detailed analysis in zero-dimensional local rings and explicit constructions. Overall, the work integrates model-theoretic purity, residue-field theory, and algebro-geometric localization to advance understanding of when smashing localizations arise from finite-type data, with significant consequences for derived categories of schemes and local rings.
Abstract
For a big tt-category, we give a characterization of the Telescope Conjecture (TC) in terms of definable tensor-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 quasi-compact quasi-separated scheme, strengthening a result \cite{HHZ21}. 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.