DISCRETE VALUATIONS CENTERED ON LOCAL DOMAINSReinhold

We study applications of discrete valuations to ideals in analytically irreducible domains, in particular applications to zero divisors modulo powers of ideals. We prove a uniform version of Izumi's theorem and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-beredness, and a valuative criterion for when the normal cone of an ideal in an integrally closed domain is reduced. Valuations of elds and eld extensions play an important role in the study of algebras and algebraic varieties. The valuations of a function eld of transcendence degree one, for instance, completely determine a smooth projective curve, giving a model of the function eld. In higher dimensions the picture is much more complicated. We present here some higher dimensional ideal-and ring-theoretic properties determined by discrete valuations centered on local domains. Most of the results in this paper are in the spirit of Rees valuations and are about the information that discrete valuations contain about powers of ideals. Section 1 explores relations among valuations, bounding one with respect to nitely and even innnitely many others either locally or globally. An example is Izumi's theorem (cf. Iz]), which was given in an algebraic setting rst by Rees R 2 ], and is given in a somewhat more general version in this paper in Theorem 1.3. Rees' proof was based on Lipman's theory of intersection multiplicities Li], and our proof uses Cutkosky's C2] results on condition (E) of Heinzer and Lantz. Section 2 treats zero divisors modulo powers of an ideal via valuations: if a product of two elements lies in a high power of an ideal, in what power does at least one of the elements have to be? Good results of this form only hold in analytically unramiied rings, which gives a new criterion of analytic irreducibility. Section 2 also contains a new criterion of one-beredness.