On a generalization of test ideals

Abstract

Abstract The test ideal τ(R) of a ring R of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal τ (a t ) associated to a given ideal a with rational exponent t ≥ 0. We first prove a key lemma of this paper (Lemma 2.1), which gives a characterization of the ideal τ (a t ). As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal τ(R) . Moreover, we prove an analogue of so-called Skoda’s theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the “modified Briançon-Skoda theorem.”