In his landmark ‘00 paper, Krause gave an abstract model theory characterization of when the Telescope Conjecture (TC) holds in a compactly generated triangulated category. Restricting to the tensor-triangulated (tt) setting, the tt version of (TC) can then be translated as “every definable ideal is the orthogonal to a set of compact objects”, as explained by R. Wagstaffe. (TC) was originally formulated for the case of the stable homotopy category of spectra, where it had been a conjecture for 40 years until the announcement of the negative answer last year. Our results are motivated by the case of D(X), the derived category of a concentrated scheme X, where (TC) is a property which often holds but fails for some X.
Balmer and Favi showed that (TC) is an affine-local property on the Balmer spectrum of a big tt-category. In the present work (arXiv:2311.00601), we show that under very mild (and conjecturally vacuous) conditions, (TC) is even stalk-local in a very strong sense: For (TC) to hold, it is enough to check that each of the Balmer’s homological residue field objects generates the local tt-category over the corresponding stalk as a definable ideal.
In the case of D(X), this ties (TC) strongly with separation properties of the adic topology of the stalk rings. We apply this to recover most known examples of validity or failure of (TC) in D(X), as well as to construct some new ones. Moreover, we show that certain restriction of (TC) can be characterized in terms of pseudoflat ring epimorphisms over R, yielding a surprising example of a non-surjective pseudoflat local ring morphism.