15 problems

Let $\mathcal{S}$ be the stable homotopy category, and let $\mathsf{BL}(\mathcal{S})$ denote its Bousfield lattice. A localizing subcategory is a full subcategory closed under the…

Sphere–projective-space conjecture. Their Bousfield classes are equal:

X(n)-adjacency conjecture. The spectra $X(n)$ and $X(n+1)$ are adjacent in the Bousfield lattice: if

Bousfield-class dichotomy conjecture. Then

The Dichotomy Conjecture. Every spectrum has either a finite local or a finite acyclic.

Finite detection conjecture for I. If $E\wedge I\neq 0$, then

Finite acyclic–finite local dichotomy conjecture. Every spectrum has either a nonzero finite acyclic or a nonzero finite local.

The infinite-regular-sequence conjecture. The Bousfield class $A^2\langle H\mathbf{F}_p\rangle$ should be the wedge of the classes $\langle S/J\rangle$ over all such infinite regul…

The decomposition conjecture for D. One has

Join-preservation conjecture. The map $r$ preserves arbitrary joins.

The minimality conjecture for A(n). If $n\geq 2$, then $\langle A(n)\rangle$ is a minimal nonzero Bousfield class. Moreover, every such $E$ is Bousfield equivalent to a finite wedg…

The quotient-isomorphism conjecture. The epimorphism $r'$ is an isomorphism.

Let $\mathcal{S}$ be the stable homotopy category, let $I$ be the Brown–Comenetz dual of the sphere spectrum, and let $\mathsf{BL}(\mathcal{S})$ be its Bousfield lattice. Hovey–Pal…

Let $\mathsf T$ be the triangulated category under consideration, and let $X$ be an object of $\mathsf T$. A compact object $C$ is $X$-acyclic if $X\otimes C=0$, and it is $X$-loca…

Retract conjecture. The map $r'$ is an isomorphism. This conjecture concerns the structure of the Bousfield lattice and its distributive sublattice; its resolution is not specified…