Grothendieck's Localization Problem

. The singularity of a fiber of a flat homomorphism of noetherian rings ' : R ! S at a prime ideal p 2 Spec R is controlled by the singularity of the fiber of ' at any specialization of p and by the singularities of the formal fibers of R at p. Introduction Let ' : R ! S be a homomorphism of commutative rings. For a prime ideal p 2 Spec R, the residue field of the localization of R at p is denoted k(p), and the ring k(p)\Omega R S is called the fiber of ' at p. The fibers of the canonical maps from R to its completions in the p--adic topologies are known as the formal fibers of R. For various properties P of commutative rings, Grothendieck [9, (7.5)] considers the following Localization Problem: Suppose that ' is a flat homomorphism of noetherian rings and that the formal fibers of R have P ; if the fibers of ' at the primes of R contracted from maximal ideals of S have P , is it then true that all the fibers of ' have P ? We obtain positive answers for properties related to "comp..