A generalization of Ohkawa's theorem (Q2877506)

In the homotopy category of spectra, one can consider the Bousfield class of a spectrum \(E\), which consists of all spectra \(X\) such that \(E\wedge X=0\); it can also be described as the \(E_*\)-spectra, where \(E_*\) is the reduced homology theory associated to \(E\). A result of Ohkawa says that the collection of all possible Bousfield classes in the homotopy category of spectra forms a set, rather than a proper class. The question of whether Bousfield classes form a set in the derived category of a commutative ring was shown in certain cases by Neeman and by Dwyer-Palmieri, and eventually was established for the general case by Stevenson and Iyengar-Krause.NEWLINENEWLINEIn this paper, the authors look for a common framework for both the homotopy category of spectra and the derived category of a commutative ring. They prove that an analogue of Ohkawa's theorem holds in the very general case of a monoidal, combinatorial, pointed model category on which is defined a functor which behaves sufficiently like a homology theory to define Bousfield classes in the model category. In doing so, they establish several new examples, such as the homotopy category of modules over a commutative ring spectrum and the stable motivic homotopy category over a Noetherian scheme of finite Krull dimension, for which the Bousfield classes form a set.