Seminar

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Let A -> 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 -> B is Cohen-Macaulay and if A is of equal characteristic zero, then all the fibers of A -> B 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.

	Notes 2.03 MB application/pdf
	Handout 248 KB application/pdf

Grothendieck's Localization Problem