Rees Valuations

This expository paper contains history, definitions, constructions, and the basic properties of Rees valuations of ideals. A section is devoted to one-fibered ideals, that is, ideals with only one Rees valuation. Cutkosky [5] proved that there exists a two-dimensional complete Noetherian local integrally closed domain in which no zero-dimensional ideal is one-fibered. However, no concrete ring of this form has been found. An emphasis in this paper is on bounding the number of Rees valuations of ideals. The last section is about the Izumi–Rees Theorem, which establishes comparability of Rees valuations with the same center. More on Rees valuations can be done via the projective equivalence of ideals, and there have been many articles along that line. See the latest article by Heinzer, Ratliff, and Rush [11], in this volume. All rings in this paper are commutative with identity, and most are Noetherian domains. The following notation will be used throughout: Q(R) denotes the field of fractions of a domain R. For any prime ideal P in a ring R, κ(P ) denotes the field of fractions of R/P . If V is a valuation ring, mV denotes its unique maximal ideal, and v denotes an element of the equivalence class of valuations naturally determined by V . We say that a Noetherian valuation is normalized if its value group is a subset of Z whose greatest common divisor is 1. If R is a ring and V is a valuation overring, then the center of V on R is mV ∩R. A valuation ring V (or a corresponding valuation v) is said to be divisorial with respect to a subdomain R if Q(R) = Q(V ) and if tr.degκ(p)κ(mV ) = Irena Swanson Department of Mathematics, Reed College, 3203 SE Woodstock Blvd., Portland, OR 97202, USA, e-mail: [email protected]