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Let f Y → 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 aarXiv:2004.06737v2 fatcat:3kwsw6ikgff35pbukmhoihbmry