TITLE: Flatly compactifiable morphisms
QUESTION [7 upvotes]: Let $f:U \to S$ be a flat morphism. Let us say that $f$ is  flatly compactifiable if there exists a proper morphism $\bar{f}:X \to S$ and a closed subscheme $Z \subset X$ such that 
1) $U = X \setminus Z$ and $f = \bar{f}_{|U}$;
2) $\bar{f}$ is proper;
3) BOTH $X$ and $Z$ are flat over $S$.
My question is whether this notion already appeared in the literature and what is the correct name for it?

REPLY [2 votes]: In general you can not expect that $f$ will be flat on $X-U$, but you have a locally finite stratification of $X$ for which $U$ is the dense stratum, and such that $f$ is flat over each strata. This notion is called "platification" (in French) and it was dealed with great details in the paper of Gruson and Raynaud "Techniques de platification d'un module".