On a generalization of test ideals

Abstract

The test ideal $\tau (R)$ 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 $t\ge 0$. We first prove a key lemma of this paper, 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 $\tau (R)$. Moreover, we prove an analog of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Briançon--Skoda theorem."

11 pages, AMS-LaTeX; v.2: minor changes, to appear in Nagoya Math. J