INTEGRALLY CLOSED FINITE-COLENGTH IDEALS IN TWO-DIMENSIONAL REGULAR LOCAL RINGS ARE

MULTIPLIER IDEALS

JOSEPH LIPMAN KEI-ICHI WATANABE

Abstract. Multiplier ideals in commutative rings are certain integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? In this note we show that in a two-dimensional regular local ring with algebraically closed residue field, in the class of ideals containing a power of the maximal ideal there is in fact no difference between ”multiplier” and ”integrally closed” (or ”complete.”) However, among ”integral” multiplier ideals (or ”adjoint ideals”) the only simple complete ideals are those of order one.