Nagoya Mathematical Journal

On a generalization of test ideals

Nobuo Hara and Shunsuke Takagi

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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 $\tau({\frak a}^t)$ associated to a given ideal $\frak a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper (Lemma \ref{key lemma}), which gives a characterization of the ideal $\tau({\frak a}^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an 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."

Article information

Nagoya Math. J., Volume 175 (2004), 59-74.

First available in Project Euclid: 27 April 2005

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Zentralblatt MATH identifier

Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]


Hara, Nobuo; Takagi, Shunsuke. On a generalization of test ideals. Nagoya Math. J. 175 (2004), 59--74.

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  • I. Aberbach and C. Huneke, An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras , Math. Ann., 297 (1993), 67–114.
  • I. Aberbach and B. MacCrimmon, Some results on test elements , Proc. Edinburgh Math. Soc., (2) 42 (1999), 541–549.
  • A. Bravo and K.E. Smith, Behavior of test ideals under smooth and étale homomorphisms , J. Algebra, 247 (2002), 78–94.
  • J. Briançon and H. Skoda, Sur la clôture intégrale dún idéal de germes de fonctions holomorphes en un point de $C^n$ , C. R. Acad. Sci. Paris Sér. A, 278 (1974), 949–951.
  • J.-P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals , Michigan Math. J., 48 (2000), 137–156.
  • N. Hara and K.-i. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities , J. Algebraic Geom., 11 (2002), 363–392.
  • N. Hara and K. Yoshida, A generalization of tight closure and multiplier ideals , Trans. Amer. Math. Soc., 355 (2003), 3143–3174.
  • M. Hochster and C. Huneke, Tight Closure and strong F-regularity , Mem. Soc. Math. France, 38 (1989), 119–133.
  • ––––, Tight closure, invariant theory and the Briançon-Skoda theorem , J. Amer. Math. Soc., 3 (1990), 31–116.
  • ––––, F-regularity, test elements, and smooth base change , Trans. Amer. Math. Soc., 346 (1994), 1–62.
  • E. Hyry, Coefficient ideals and the Cohen-Macaulay property of Rees algebras , Proc. Amer. Math. Soc., 129 (2001), 1299–1308.
  • R. Lazarsfeld, Positivity in algebraic geometry, book, to appear.
  • J. Lipman, Adjoints of ideals in regular local rings , Math. Research Letters, 1 (1994), 739–755.
  • G. Lyubeznik and K.E. Smith paper Strong and weak F-regularity are equivalent for graded rings, Amer. J. Math., 121 (1999), 1279–1290.
  • ––––, On the commutation of the test ideal under localization and completion , Trans. Amer. Math. Soc., 353 (2001), no. 8, 3149–3180.
  • D.G. Northcott and D. Rees, Reductions of ideals in local rings , Proc. Camb. Philos. Soc., 50 (1954), 145–158.
  • S. Takagi, An interpretation of multiplier ideals via tight closure , J. Algebraic Geom., 13 (2004), 393–415.
  • K.-i. Watanabe, F-regular and F-pure normal graded rings , J. Pure Appl. Algebra, 71 (1991), 341–350.