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The multiplier ideal is a universal test ideal

Two different ideals have been defined during the last decade, each of which has played an important role in its respective area. In commutative algebra, the test ideal defined by Hochster and Huneke [HHl] plays a central role in the theory of tight closure. In algebraic geometry, multiplier ideals, defined by Nadel [Na] and reinterpreted by others (see, e.g. [Ei]), have refined various vanishing theorems and have had interesting applications, for example, to the existence of Kahler-Einstein metrics on certain Fano varieties. These two different types of ideals have arisen independently, are defined differently, and are motivated by different concerns. The purpose of this note is to show that, in fact, test ideals and multiplier ideals are closely related. With recent advances in commutative algebra, tying tight closure to singularity theory in algebraic geometry [W2], [S2], [Ha2], came good reason to suspect a connection between test ideals and multiplier ideals. The test ideal 7 of a commutative ring R defines, a t least conjecturally, the locus of non-F-regular points in Spec Rin other words, a prime idea! P contains r if and only if some ideal of the local ring Rp is not tightly closed. Likewise, the multiplier ideal N of R defines the locus of non-log-terminal points of Spec R. Because F-regularity is equivalent to logterminal singularities (for QGorenstein rings of characteristic zero), the test ideal and the multiplier ideal must have the same radical, and it is natural to investigate whether they are indeed the same ideal. Theorem 3.1 in this paper establishes, roughly speaking, that the test ideal and the multiplier ideal are indeed equal for every normal local QGorenstein ring