4 problems

Localizing–colocalizing correspondence conjecture. There is a one-to-one correspondence between localizing subcategories and colocalizing subcategories. Consequently, assuming the…

Principal-localizing-subcategory conjecture. Every localizing subcategory is the collection of $E$-acyclics for some spectrum $E$, and is therefore principal.

Let $X$ be a noetherian scheme. A coherent subset of $X$ is a subset of the underlying topological space of the kind appearing in Krause's classification for commutative noetherian…

Bousfield's conjecture. The localizing subcategory $\mathcal{L}(F)$ has a set of generators.