A uniform treatment of Grothendieck's localization problem
Abstract
Let $f\colon Y \to X$ be a proper flat morphism of locally noetherian schemes. Then, the locus in $X$ over which $f$ is smooth is stable under generization. We prove that under suitable assumptions on the formal fibers of $X$, the same property holds for other local properties of morphisms, even if $f$ is only closed and flat. Our proof of this statement reduces to a purely local question known as Grothendieck's localization problem. To answer Grothendieck's problem, we provide a general framework that gives a uniform treatment of previously known cases of this problem, and also solves this problem in new cases, namely for weak normality, seminormality, $F$rationality, and the property "CohenMacaulay and $F$injective." For the weak normality statement, we prove that weak normality always lifts from Cartier divisors. We also solve Grothendieck's localization problem for terminal, canonical, and rational singularities in equal characteristic zero.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.06737
 Bibcode:
 2020arXiv200406737M
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Commutative Algebra;
 14B07 (Primary) 13H10;
 14B25;
 13J10;
 13F45;
 13A35 (Secondary)
 EPrint:
 29 pages. v2: Fixed proof that weak normality lifts, added results on terminal and canonical singularities, other small changes