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 valuation ring, $X$ and $S$ two $R$-flat and universally closed $R$-schemes.

Let $f : X\to S$ be a universally closed morphism. Assume the special fiber of $f$ is an isomorphism.

- If $X$ and $S$ are reduced, is $f$ an isomorphism?
- If $X$ and $S$ have reduced fibers, is $f$ an isomorphism?

Without either of the reducedness assumptions, the answer is "no", as shown by the example provided here.

My expectation is that the answer is still "no" to both questions, but an example seems subtle to produce.

**Remarks**
In these remarks we further assume $f$ is flat, to see what more can be said in this case. After all, should the questions have positive answer, $f$ would have to be flat. The remarks are listed increasing the strength of the additional assumptions, to see what can be said in each ever-more-special case.

From flatness we deduce $f$ is surjective, since generizations lift along $f$ (by flatness), and every point in $S$ specializes to a point in the special fiber (by universal closedness of $S$, if $s\in S$ is contained in the generic fiber, it maps to the generic point in $\text{Spec}(R)$. This specializes to the closed point in $\text{Spec}(R)$, and specializations lift along $S\to \text{Spec}(R)$ by universal closedness, so there is some $s_0\in S$ contained in the special fiber, to which $s$ specializes).

If $f$ is, in addition,

*separated*and*locally of finite type*, then it is also proper, since $f$ is quasi-compact by universal closedness. It is therefore an isomorphism (it is quasi-finite, hence finite, hence finite flat of degree $1$). The question is interesting only if $f$ is*not*locally of finite type.To summarize, the case when $f$ is flat and separated is already unanswered but one has access to more info on $f$. A non-separated example, flat or not, to answer the question in the negative, would be already what I'm looking for. A flat separated example, if any, would be the best way to settle this puzzle, and I expect it is very subtle to find.