A uniform treatment of Grothendieck's localization problem

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 stable under generization. We prove that, suitable assumptions on formal fibers , same property holds for other local properties morphisms, even if only closed and flat. Our proof this statement reduces to purely question known as Grothendieck's localization problem. To answer problem, we provide general framework that gives uniform treatment previously cases also solves problem new cases, namely weak normality, seminormality, $F$ -rationality, ‘Cohen–Macaulay -injective’ property. For normality statement, always lifts from Cartier divisors. solve terminal, canonical, rational singularities equal characteristic zero.