# A uniform treatment of Grothendieck's localization problem

@article{Murayama2022AUT,
title={A uniform treatment of Grothendieck's localization problem},
author={Takumi Murayama},
journal={Compositio Mathematica},
year={2022},
volume={158},
pages={57 - 88}
}
• T. Murayama
• Published 14 April 2020
• Mathematics
• Compositio Mathematica
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…
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