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For a big tt-category, we give a characterization of the Telescope Conjecture(TC) in terms of definable tensor-ideals generated by homological residuefields. We formulate a stalk-locality property of (TC) and prove that it holdsin the case of the derived category of a quasi-compact quasi-separated scheme,strengthening a result \cite{HHZ21}. As an application, we find strong linksbetween (TC) and separation properties of the adic topology on local rings.This allows us to recover known examples and counterexamples of when (TC) holdsover a scheme, as well as to construct some new ones.

Telescope conjecture via homological residue fields with applications to schemes