Let $X$ be a scheme over $S$. Suppose $D$ and $D^\prime$ are two closed subschemes of $X$ which are both proper over $S$. So is the scheme-theoretic sum of $D$ and $D^\prime$ again proper over $S$.
If the above does not hold generally, let us suppose moreover that $D$ and $D^\prime$ are both effective Cartier Divisors, i.e., both are locally principal closed subschemes. In this case is the sum of the two divisors proper over $S$?
Well, recently I am reading Katz and Mazur's book $Arithmetic \space Moduli \space of \space Elliptic \space Curves$ and the question arises in the study of effective Cartier divisors of smooth curves over S. If you have any other advice in studying this topic, please point out directly. I will be grateful!
Sincerely.
This is just an expansion of my comment above. All intersections and unions are scheme-theoretic.
One has $f:Y=D\cup D'\to S$, where $D,D'$ are closed subschemes of $Y$. One wants to show $f$ is proper if $f_{|D}, f_{|D'}$ are proper. Proper just means universally closed. So, take a morphism $T\to S$ and pull back to get $g:Y\times_S T=D\times_S T\cup D'\times_S T\to T$. If $Z\subset Y\times_S T$ is a closed subset, then $Z=(Z\cap D\times_S T)\cup (Z\cap D'\times_S T)$ and $g(Z)=g_{|D\times_S t}(Z\cap D\times_S T)\cup g_{|D'\times_S T}(Z\cap D'\times_S T)$. Since both the right hand terms are closed in $T$, since by assumption, you are done.