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In this question I'd like to examine some properties of universally closed morphisms. The question is self-contained. It can also be seen as a follow-up to this question. Let $R$ be a discrete ...

Let $R$ be a dvr and $f : X\to S$ a universally closed morphism of $R$-schemes. Assume $X$ and $S$ are $R$-flat and universally closed. If the special fiber of $X\to S$ is a closed immersion, is $X\...