On a generalization of test ideals

Details On a generalization of test ideals On a generalization of test ideals

2002-10-09

arxiv.org/abs/math/0210131v2

Mathematics

Commutative Algebra

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

Scientific paper

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 $\tau(\a^t)$ 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 $\tau(\a^t)$. 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 Brian\c{c}on--Skoda theorem."

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