4 problems

Affine-line failure conjecture. The analogous agreement already fails for the affine line $\mathbb Z[x]$; in particular, the displayed spectra are not homeomorphic. The claim conce…

Let $A$ and $B_i$ be homotopical rings, and let $\{f_i:A\to B_i\}_{i\in I}$ be a family of homotopical epimorphisms. Suppose that the family is a cover for the fine Zariski topolog…

Let $G$ be an infinite group in the class ${\mathscr C}$ of groups with cofinite Zariski topology. Main conjecture. The group $G$ is necessarily abelian; consequently, every infini…

Let $G$ be a group with cofinite Zariski topology. Cofinite Zariski conjecture. The group $G$ is necessarily either abelian or finite. The conjecture is motivated by the strong res…