Strict localization of cohomological Bousfield classes

Determine whether every cohomological Bousfield class in an arbitrary well-generated tensor-triangulated category is strictly localizing.

Background

Homological Bousfield classes are kernels of tensor-localization functors and are therefore strictly localizing. The paper points out that the analogous property for cohomological Bousfield classes is not established in general. Resolving this question would clarify whether all cohomological annihilator ideals arise as kernels of Bousfield localizations.

References

In contrast, it is not known whether every cohomological Bousfield class ${A}$ is strictly localizing, nor is it known in general whether there is only a set of cohomological Bousfield classes.

— Local Bousfield classes via homological support  (2608.26876 - Barthel et al., 27 Aug 2026) in Remark following Example 2.1, Section 2, “Homological and cohomological Bousfield classes”