Local Bousfield classes via homological support

Abstract

Given an object $A$ in a big tensor-triangulated category, we study the homological and cohomological Bousfield classes of the associated localization: the tensor-triangulated category of $A$-local objects. We show that the homological support classifies the homological Bousfield classes of the $A$-local category precisely when an $A$-relative form of the homological detection property holds. Moreover, we prove that this holds if and only if $A$ is Bousfield equivalent to a coproduct of homological residue fields. The analogous classification of cohomological Bousfield classes by homological cosupport is strictly stronger: it is equivalent to an $A$-relative form of homological stratification. This equivalence between stratification and the classification of cohomological Bousfield classes is new even in the absolute case. A further surprise is that stratification is also equivalent to the classification of homological Bousfield classes together with the statement that every cohomological Bousfield class is homological. Applied to chromatic homotopy theory, these results classify the homological Bousfield classes of any localization of spectra with respect to a coproduct of Morava $K$-theories. This covers many localizations of interest. We also completely characterize when such chromatic localizations are relatively homologically stratified. This yields new examples of cohomological Bousfield classes that are not homological. In particular, it answers a question of Wolcott concerning the category of harmonic spectra. Our examples are produced by exhibiting local spectra with empty homological cosupport.