Integrally Closed Ideals in Two-dimensional Regular Local Rings Are Multiplier Ideals

Multiplier ideals in commutative rings are certain integrally closed ideals with properties that lend themselves to highly interesting applications. How special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field there is in fact no difference between “multiplier” and “integrally closed” (or “complete.”) But among multiplier ideals arising from an integer multiplying constant (also known as adjoint ideals), and primary for the maximal ideal, the only simple complete ideals are those of order one. Introduction. There has arisen in recent years a substantial body of work on multiplier ideals in commutative rings (see [La]). Multiplier ideals are integrally closed ideals with properties that lend themselves to highly interesting applications. One is tempted then to ask just how special multiplier ideals are among integrally closed ideals in general. In this note we show that in a two-dimensional regular local ring R with maximal ideal m such that the residue field R/m is algebraically closed there is actually no difference between multiplier ideals and integrally closed ideals. In fact it turns out more convenient to do this for fractionary R-ideals, i.e., nonzero finitely-generated R-submodules of the fraction field L of R. Main Result. Every integrally closed fractionary R-ideal is a multiplier ideal. After this paper was first submitted, we learned that independently of us C. Favre and M. Jonsson had found a related proof [FJ, §6]. Their argument is given in the context of a novel treatment of valuations of R. Though we thought initially that our proof applied only to m-primary ideals, Favre and Jonnson had no such restriction. This prompted us to reexamine our proof, which we then found could be made to apply to the general case as well. 2000 Mathematics Subject Classification. 13B22, 13H05. First author partially supported by the National Security Agency. Second author partially supported by Grants-in-Aid in Scientific Researches, 13440015, 13874006; and his stay at MSRI was supported by the Bunri Fund, Nihon University. Both authors are grateful to MSRI for providing the environment without which this work would not have begun. Research at MSRI is supported in part by NSF grant DMS-9810361. It most likely suffices that R/m be infinite, but we want to avoid additional technicalities.