Telescope conjecture via homological residue fields with applications to schemes

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.