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On a generalization of test ideals

Published online by Cambridge University Press:  22 January 2016

Nobuo Hara and
Shunsuke Takagi
Nobuo Hara
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan, hara@math.tohoku.ac.jp
Shunsuke Takagi
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro Tokyo 153-8914, Japan, stakagi@ms.u-tokyo.ac.jp
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Abstract

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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(at) 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(at). 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.”

Keywords

13A35

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 175 , 2004, pp. 59 - 74
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

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