Hovey and Hovey–Palmieri conjectures for spectra

Prove or refute, in the stable homotopy category of spectra, both that every cohomological Bousfield class is homological and that every localizing ideal is a homological Bousfield class.

Background

The paper recalls two conjectures concerning the stable homotopy category. Hovey conjectured that every cohomological Bousfield class is homological, while Hovey and Palmieri proposed the stronger assertion that every localizing ideal is a homological Bousfield class. The authors show that these assertions fail for general tensor-triangulated categories, but explicitly state that they remain unresolved for the stable homotopy category.

References

These conjectures remain open for $$ but we will provide many examples showing that they are false for general tensor-triangulated categories~$\cat T$.

— Local Bousfield classes via homological support  (2608.26876 - Barthel et al., 27 Aug 2026) in Remark 2.3, Section 2, “Homological and cohomological Bousfield classes”