 
Summary: GROTHENDIECK'S LOCALIZATION PROBLEM
Luchezar L. Avramov and Hans{Bjrn Foxby
Abstract. The singularity of a ber of a
at homomorphism of noetherian rings ' : R ! S at
a prime ideal p 2 Spec R is controlled by the singularity of the ber of ' at any specialization
of p and by the singularities of the formal bers of R at p.
Introduction
Let ' : R ! S be a homomorphism of commutative rings. For a prime ideal p 2 Spec R,
the residue eld of the localization of R at p is denoted k(p), and the ring
k(p)
R S is
called the ber of ' at p. The bers of the canonical maps from R to its completions in
the p{adic topologies are known as the formal bers of R.
For various properties P of commutative rings, Grothendieck [9, (7.5)] considers the
following Localization Problem: Suppose that ' is a
at homomorphism of noetherian rings
and that the formal bers of R have P ; if the bers of ' at the primes of R contracted
from maximal ideals of S have P , is it then true that all the bers of ' have P ? We
obtain positive answers for properties related to \complete intersection," \Gorenstein," and
\Cohen{Macaulay" in Section 4 and compare them to earlier ones in Section 5. However,
the main thrust of this paper is to investigate the more general thesis formulated in the
Abstract.
