A uniform treatment of Grothendieck's localization problem

  title={A uniform treatment of Grothendieck's localization problem},
  author={Takumi Murayama},
  journal={Compositio Mathematica},
  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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