# Grothendieck's localization problem

## Location

Let f : Y -->X be a proper flat morphism of algebraic varieties. Grothendieck and Dieudonné showed that the smoothness of f can be detected at closed points of X. Using André–Quillen homology, André showed that when X is excellent, the same conclusion holds when f is a closed flat morphism between locally noetherian schemes. We give a new proof of André's result using a version of resolutions of singularities due to Gabber. Our method gives a uniform treatment of Grothendieck's localization problem and resolves various new cases of this problem, which asks whether similar statements hold for other local properties of morphisms.

The discussion for Takumi Murayama’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2021-01-22-tm (and will be deleted after ~3-7 days).