Are all localizing subcategories of a stable homotopy category coreflexive?

J. Rosický¹

Masaryk University, Brno

Localizing subcategories are very important in stable homotopy theory and there is not known any example of a localizing subcategory $\mathcal L$ without a localization functor, which is the same thing as $\mathcal L$ not being coreflective. We will confirm the suspicion of M. Hovey, J. H. Palmieri and N. P. Strickland that the answer may depend on set theory by showing that, assuming Vopěnka's principle, every localizing subcategory $\mathcal L$ of the homotopy category $\mathcal S$ of spectra is coreflective. Moreover, $\mathcal L$ is generated by a single object and, dually, every colocalizing subcategory $\mathcal C$ of $\mathcal S$ is reflective and generated by a single object. The consequence is that every localizing subcategory of $\mathcal S$ is a cohomological Bousfield class.

Footnotes

...ý¹

joint work with C. Casacuberta and J. Gutiérrez