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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, [email protected]
Shunsuke Takagi
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro Tokyo 153-8914, Japan, [email protected]
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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
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 175 , 2004 , pp. 59 - 74
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2004

References

[AH] Aberbach, I. and Huneke, C., An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras, Math. Ann., 297 (1993), 67–114.Google Scholar
[AM] Aberbach, I. and MacCrimmon, B., Some results on test elements, Proc. Edinburgh Math. Soc., (2) 42 (1999), 541–549.Google Scholar
[BSm] Bravo, A. and Smith, K. E., Behavior of test ideals under smooth and étale homo-morphisms, J. Algebra, 247 (2002), 78–94.Google Scholar
[BSk] Briançon, J. and Skoda, H., Sur la cloture intégrale dun idéal de germes de fonctions holomorphes en un point de C”, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 949–951.Google Scholar
[DEL] Demailly, J.-P., Ein, L. and Lazarsfeld, R., A subadditivity property of multiplier ideals, Michigan Math. J., 48 (2000), 137–156.CrossRefGoogle Scholar
[HW] Hara, N. and Watanabe, K.-i., F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom., 11 (2002), 363–392.Google Scholar
[HY] Hara, N. and Yoshida, K., A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc., 355 (2003), 3143–3174.Google Scholar
[HH1] Hochster, M. and Huneke, C., Tight Closure and strong F-regularity, Mem. Soc. Math. France, 38 (1989), 119–133.Google Scholar
[HH2] Hochster, M. and Huneke, C., Tight closure, invariant theory and the Briançon-Skoda theorem, J. Amer. Math. Soc., 3 (1990), 31–116.Google Scholar
[HH3] Hochster, M. and Huneke, C., F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994), 1–62.Google Scholar
[Hy] Hyry, E., Coefficient ideals and the Cohen-Macaulay property of Rees algebras, Proc. Amer. Math. Soc., 129 (2001), 1299–1308.Google Scholar
[La] Lazarsfeld, R., Positivity in algebraic geometry, book, to appear.Google Scholar
[Li] Lipman, J., Adjoints of ideals in regular local rings, Math. Research Letters, 1 (1994), 739–755.Google Scholar
[LS1] Lyubeznik, G. and Smith, K.E paper Strong and weak F-regularity are equivalent for graded rings, Amer. J. Math., 121 (1999), 1279–1290.Google Scholar
[LS2] Lyubeznik, G. and Smith, K.E, On the commutation of the test ideal under localization and completion, Trans. Amer. Math. Soc., 353 (2001), no. 8, 3149–3180.Google Scholar
[NR] Northcott, D.G. and Rees, D., Reductions of ideals in local rings, Proc. Camb. Philos. Soc., 50 (1954), 145–158.Google Scholar
[Ta] Takagi, S., An interpretation of multiplier ideals via tight closure, J. Algebraic Geom., 13 (2004), 393–415.Google Scholar
[W] Watanabe, K.-i., F-regular and F-pure normal graded rings, J. Pure Appl. Algebra, 71 (1991), 341–350.Google Scholar