I am interested in a morphism of $S$-schemes $f : X \to Y$ such that $X$ and $Y$ are proper over $S$ and there is some $s\_0 \in S$ such that $f : X\_{s\_0} \to Y\_{s\_0}$ is a closed immersion. Is it true that there is an open neighborhood $U \subset S$ of $s\_0$ so that $X\_U \to Y\_U$ is a closed immersion?


Because $f$ is automatically proper, I hoped to use the characterization of closed immersions as proper monomorphisms. Using an argument on formal neighborhoods of the fibers, I think I can show that $f : X \to Y$ is unramified at each point of the fiber and hence should be unramified on some neighborhood of the fiber. Maybe I can also show that $f$ is radicial in a similar fashion?


I am okay with assuming that $X, Y$ are noetherian and flat over $S$ if this helps.



Proper morphisms that are closed immersion on a fiber
This is proved in [EGA III, tome 1, Proposition 4.6.7(i)](http://www.numdam.org/item/PMIHES_1961__11__5_0/).


