2 problems

Let $K$ be a global field, let $\mathcal{O}_S$ be a regular semilocal subring of $K$, and set $\mathcal{S}=\operatorname{Spec}(\mathcal{O}_S)$. Let $\mathcal{X}$ be a regular schem…

Let $K$ be a local field with ring of integers $\mathcal{O}_K$ and finite residue field $k$. Let $X$ be a regular proper flat scheme over $\mathcal{O}_K$, with smooth generic fiber…