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A generalization of tight closure and multiplier ideals


Authors: Nobuo Hara and Ken-ichi Yoshida
Journal: Trans. Amer. Math. Soc. 355 (2003), 3143-3174
MSC (2000): Primary 13A35, 14B05
DOI: https://doi.org/10.1090/S0002-9947-03-03285-9
Published electronically: April 11, 2003
MathSciNet review: 1974679
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Abstract: We introduce a new variant of tight closure associated to any fixed ideal $\mathfrak{a}$, which we call $\mathfrak{a}$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau (\mathfrak{a})$ of all $\mathfrak{a}$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $\tau (\mathfrak{a})$ and the multiplier ideal associated to $\mathfrak{a}$ (or, the adjoint of $\mathfrak{a}$ in Lipman's sense) in normal $\mathbb{Q} $-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau (\mathfrak{a})$ similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau (\mathfrak{a})$ and the F-rationality of Rees algebras.


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  • [AKM] I. Aberbach, M. Katzman, and B. MacCrimmon, Weak F-regularity deforms in $\mathbb Q $-Gorenstein rings, J. Algebra 204 (1998), 281-285. MR 99d:13003
  • [AM] I. Aberbach and B. MacCrimmon, Some results on test ideals, Proc. Edinburgh Math. Soc. (2) 42 (1999), 541-549. MR 2000i:13005
  • [B] J.-F. Boutot, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math. 88 (1987), 65-68. MR 88a:14005
  • [BS] H. Skoda and J. Briançon, Sur la cl$\hat {o}$ture intégrale d'un idéal de germes de fonctions holomorphes en un point de $C^{n}$, C. R. Acad. Sci. Paris Sér. A 278 (1974), 949-951. MR 49:5394
  • [BH] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 95h:13020
  • [DEL] J.-P. Demailly, L. Ein and R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137-156. MR 2002a:14016
  • [Ei] L. Ein, Multiplier ideals, vanishing theorems and applications, in Algebraic Geometry--Santa Cruz 1995, pp. 203-219, Proc. Sympos. Pure Math., vol. 62, American Mathematical Society, Providence, RI, 1997. MR 98m:14006
  • [ELS] L. Ein, R. Lazarsfeld and K. E. Smith, Uniform bounds and symbolic powers on smooth varieties, Invent. Math. 144 (2001), 241-252. MR 2002b:13001
  • [FW] R. Fedder and K.-i. Watanabe, A characterization of F-regularity in terms of F-purity, in Commutative Algebra, Berkeley 1987, pp. 227-245, Math. Sci. Res. Inst. Publ., vol. 15, Springer-Verlag, New York, 1989. MR 91k:13009
  • [GS] S. Goto and Y. Shimoda, On the Rees algebras of Cohen-Macaulay local rings, in Commutative Algebra, Fairfax 1979, pp. 201-231, Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982. MR 84a:13021
  • [GW] S. Goto and K.-i. Watanabe, On graded rings, I, J. Math. Soc. Japan 30 (1978), 179-213. MR 81m:13021
  • [Ha1] N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981-996. MR 99h:13005
  • [Ha2] -, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885-1906. MR 2001m:13009
  • [HT] N. Hara and S. Takagi, Some remarks on a generalization of test ideals, preprint.
  • [HW] N. Hara and K.-i. Watanabe, F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geometry 11 (2002), 363-392. MR 2002k:13009
  • [HWY1] N. Hara, K.-i. Watanabe, and K. Yoshida, F-rationality of Rees algebras, J. Algebra 247 (2002), 153-190.
  • [HWY2] -, Rees algebras of F-regular type, J. Algebra 247 (2002), 191-218.
  • [HHK] M. Herrmann, E. Hyry and T. Korb, On Rees algebras with a Gorenstein Veronese subring, J. Algebra 200 (1998), 279-311. MR 98m:13006
  • [Ho1] M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488. MR 57:3111
  • [Ho2] M. Hochster, The tight integral closure of a set of ideals, J. Algebra 230 (2000), 184-203. MR 2002f:13009
  • [HH0] M. Hochster and C. Huneke, Tight closure and strong F-regularity, Mem. Soc. Math. France 38 (1989), 119-133. MR 91i:13025
  • [HH1] -, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31-116. MR 91g:13010
  • [HH2] -, F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1-62. MR 95d:13007
  • [HH3] -, Tight closure in equal characteristic zero, to appear.
  • [How] J. A. Howald, Multiplier ideals of monomial ideals, Trans. Amer. Math. Soc. 353 (2001), 2665-2671. MR 2002b:14061
  • [Hu] C. Huneke, Tight closure and its applications, C.B.M.S. Regional Conference Series in Mathematics, No. 88, American Mathematical Society, Providence, RI, 1996. MR 96m:13001
  • [Hy1] E. Hyry, Blow-up rings and rational singularities, Manuscripta Math. 98 (1999), 377-390. MR 2001d:13002
  • [Hy2] -, Coefficient ideals and the Cohen-Macaulay property of Rees algebras, Proc. Amer. Math. Soc. 129 (2001), 1299-1308. MR 2001h:13005
  • [Ka] Y. Kawamata, The cone of curves of algebraic varieties, Ann. Math. 119 (1984), 603-633. MR 86c:14013b
  • [Ku] E. Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), 999-1013. MR 55:5612
  • [La] R. Lazarsfeld, Positivity in Algebraic Geometry, preprint.
  • [Li] Lipman, J., Adjoints of ideals in regular local rings, Math. Res. Letters 1 (1994), 739-755. MR 95k:13028
  • [Mc] B. MacCrimmon, Weak F-regularity is strong F-regularity for rings with isolated non-$\mathbb Q $-Gorenstein points, Trans. Amer. Math. Soc., to appear.
  • [Ma] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. MR 88h:13001
  • [MS] V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math. 1 (1997), 249-278. MR 99e:13009
  • [N] A. Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. Math. 132 (1990), 549-596. MR 92d:32038
  • [Si] A. K. Singh, F-regularity does not deform, Amer. J. Math. 121 (1999), 919-929. MR 2000e:13006
  • [Sm1] K. E. Smith, F-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159-180. MR 97k:13004
  • [Sm2] -, The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), 5915-5929. MR 2002d:13008
  • [T] S. Takagi, An interpretation of multiplier ideals via tight closure, preprint.
  • [VV] P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93-101. MR 80d:14010
  • [Vr] A. Vraciu, Strong test ideals, J. Pure Appl. Algebra 167 (2002), 361-373. MR 2003a:13004
  • [Wa] K.-i. Watanabe, F-regular and F-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341-350. MR 92g:13003
  • [Wi] L. J. Williams, Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721-743. MR 96f:13003

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Additional Information

Nobuo Hara
Affiliation: Mathematical Institute, Tohoku University, Sendai 980–8578, Japan
Email: hara@math.tohoku.ac.jp

Ken-ichi Yoshida
Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464–8602, Japan
Email: yoshida@math.nagoya-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-03-03285-9
Received by editor(s): August 20, 2002
Received by editor(s) in revised form: December 19, 2002
Published electronically: April 11, 2003
Additional Notes: Both authors are partially supported by a Grant-in-Aid for Scientific Research, Japan
Article copyright: © Copyright 2003 American Mathematical Society

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