  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.
