Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-15T00:48:50.815Z Has data issue: false hasContentIssue false
  • English
  • Français

A New Version of α-Tight Closure

Published online by Cambridge University Press:  11 January 2016

Adela Vraciu
Adela Vraciu*
Affiliation:
Department of Mathematics University of South Carolina, Columbia SC 29205, U.S.A., vraciu@math.sc.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Hara and Yoshida introduced a notion of α-tight closure in 2003, and they proved that the test ideals given by this operation correspond to multiplier ideals. However, their operation is not a true closure. The alternative operation introduced here is a true closure. Moreover, we define a joint Hilbert-Kunz multiplicity that can be used to test for membership in this closure. We study the connections between the Hara-Yoshida operation and the one introduced here, primarily from the point of view of test ideals. We also consider variants with positive real exponents.

Keywords

13A35

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 192 , 2008, pp. 1 - 25
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[Ab] Aberbach, I., Extensions of weakly and strongly F-regular rings by flat maps, J. Algebra, 241 (2001), 799–807.Google Scholar
[Bt] Bhattacharya, P. B., The Hilbert function of two ideals, Proc. Cambridge Philos. Soc, 53 (1957), 568–575.Google Scholar
[BMS] Blickle, M., Mustata, M., and Smith, K. E., Discretness and rationality of F-thresholds, preprint.Google Scholar
[ELSV] Ein, L., Lazarsfeld, R., Smith, K. E. and Varolin, D., Jumping coefficients for multiplier ideals, Duke Math. J., 123 (2004), 469–506.Google Scholar
[Ep] Epstein, N., A tight closure analogue of analytic spread, Math. Proc. Cambridge Philos. Soc, 139 (2005), no. 2, 371–383.Google Scholar
[Hn] Hanes, D., Notes on the Hilbert-Kunz function, J. Algebra, 265 (2003), no. 2, 619–630.CrossRefGoogle Scholar
[H1] Hara, N., Characterization of rational singularities in terms of the injectivity of Frobenius, Amer. J. Math., 120 (1998), no. 5, 981–996.CrossRefGoogle Scholar
[H2] Hara, N., Geometric interpretation of test ideals, Trans. Amer. Math. Soc, 353 (2001), no. 5, 1885-1906.Google Scholar
[HT] Hara, N. and Tagaki, S., On a generalization of test ideals, Nagoya Math. J., 175 (2004), 59–74.Google Scholar
[HY] Hara, N. and Yoshida, K., A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc, 355 (2003), no. 8, 3143–3174.Google Scholar
[Ho] Hochster, M., Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc, 231 (1997), no. 2, 463–488.Google Scholar
[HH1] Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançcon-Skoda theorem, J. Amer. Math. Soc, 3 (1990), 31–116.Google Scholar
[HH2] Hochster, M. and Huneke, C., Localization and test exponents for tight closure, Michigan Math. J., 48 (2000), 305–329.Google Scholar
[HS] Huneke, C. and Smith, K., Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math., 484 (1997), 127–152.Google Scholar
[Mo] Monsky, P., The Hilbert-Kunz function, Math. Ann., 263 (1983), 43–49.Google Scholar
[MTW] Mustata, M., Tagaki, S., and Watanabe, K.-i., F-thresholds and Bernstein-S ato polynomials, European Congress of Mathematics, Eur. Math. Soc., Zrich, 2005, pp. 341–364.Google Scholar
[NR] Northcott, D. G., and Rees, D., Reductions of ideals in local rings, Proc. Cambridge Philos. Soc., 50 (1954), 145–158.Google Scholar
[S1] Smith, K. E., Tight closure in graded rings, J. Math. Kyoto Univ., 37 (1997), no. 1, 35–53.Google Scholar
[S2] Smith, K. E., The multiplier ideal is a universal test ideal, Comm. Algebra, 28 (2000), 5915–5929.CrossRefGoogle Scholar
[Vr] Vraciu, A., ∗-Independence and special tight closure, J. Algebra, 249 (2002), 544–565.Google Scholar
[WY] Watanabe, K. and Yoshida, K., Hilbert-Kunz multiplicity and an inequality between multiplicity and colength, J. Algebra, 230 (2000), 295–317.Google Scholar