Grothendieck's localization problem

Abstract: Let $A\to B$ be a flat local map of noetherian complete local rings. Using Hironaka's resolution of singularities Grothendieck and Dieudonné showed that if the closed fiber of the map $A\to B$ is Cohen-Macaulay and if $A$ is of equal characteristic zero, then all the fibers of the map are Cohen-Macaulay. Three decades later, Avramov and Foxby showed that the same statement holds without the characteristic assumption on A. Grothendieck's localization problem asks whether a similar statement holds with Cohen-Macaulayness replaced by other local properties of noetherian local rings. We solve Grothendieck's localization problem for all sufficiently well-behaved properties of noetherian local rings. Our proof provides a uniform treatment of previously known special cases of Grothendieck's problem, in particular giving a new proof of Avramov and Foxby's result. As an application, we show that if the closed fibers of a flat morphism of algebraic varieties are smooth, then all fibers are smooth.

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

You have to register to attend the seminar; the registration link is on the webpage. But you only have to register once. Once you register for a seminar, your registration will be carried over (in theory!) to future seminars.