4 problems

A valuation ring is a commutative valuation ring, and a filtered colimit is a colimit indexed by a filtered category. Zariski's local uniformisation conjecture. Every valuation rin…

A chain ring is a ring whose ideals are totally ordered by inclusion; equivalently, for any two elements, one divides the other. Chain-ring quotient conjecture. Every chain ring is…

Knaf's conjecture. If

Polystable reduction conjecture. There exists a modification $Y\to X$ such that $Y_\eta=X_\eta$ and $Y$ is polystable over ${\mathcal O}$.