Flatly compactifiable morphisms - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T19:14:32Z http://mathoverflow.net/feeds/question/41809 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41809/flatly-compactifiable-morphisms Flatly compactifiable morphisms Sasha 2010-10-11T16:48:13Z 2012-03-31T11:17:46Z <p>Let $f:U \to S$ be a flat morphism. Let us say that $f$ is <b> flatly compactifiable</b> if there exists a proper morphism $\bar{f}:X \to S$ and a closed subscheme $Z \subset X$ such that </p> <p>1) $U = X \setminus Z$ and $f = \bar{f}_{|U}$;</p> <p>2) $\bar{f}$ is proper;</p> <p>3) BOTH $X$ and $Z$ are flat over $S$.</p> <p>My question is whether this notion already appeared in the literature and what is the correct name for it?</p> http://mathoverflow.net/questions/41809/flatly-compactifiable-morphisms/92744#92744 Answer by Franz for Flatly compactifiable morphisms Franz 2012-03-31T11:02:39Z 2012-03-31T11:17:46Z <p>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".</p>