# Grothendieck's localization problem

### Takumi Murayama (Princeton University)

22-Jan-2021, 20:00-21:00 (2 years ago)

Abstract: Let $f\colon Y \rightarrow X$ 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.

algebraic geometry

Audience: researchers in the topic

( paper )