5 problems

Let $S$ be a noetherian excellent finite dimensional base scheme of characteristic $0$. The Ayoub conjecture. The heart of the perverse homotopy t-structure on $\mathrm{DM}(S)$ is…

Cycle-module conjecture. The graded groups $H^*E$ form a cycle module over $X$. The conjecture asks for the Rost cycle-module structure, including its required residue and functori…

Morel's conjecture. There should be a presentation of the heart of the stable homotopy category $\mathbf{SH}(k)$, equivalently the category of homotopy modules, in terms of a non-o…

Cycle-module equivalence conjecture. The functor $H_{cm}$ yields an equivalence of $\underline{H}_{t_{hom}}(S)$ with a certain abelian category of cycle modules.

Cycle-module conjecture. The heart of $t_{hom}(S)$ is given by cycle modules over $S$.