Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-31T04:43:24.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results on test elements

Published online by Cambridge University Press:  20 January 2009

Ian M. Aberbach  and
Brian Maccrimmon
Ian M. Aberbach
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A., E-mail address: aberbach@math.missouri.edu
Brian Maccrimmon
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A., E-mail address: aberbach@math.missouri.edu
Rights & Permissions [Opens in a new window]

Extract

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.

Let (R, m) be a Noetherian local reduced ring of positive prime characteristic. We show that if R is ℚ-Gorenstein then the test ideal of R localizes, which extends a result of K. E. Smith. We also show that if Rc, is weakly F-regular and ℚ-Gorenstein, then c has a power which is a completely stable test element. This extends results of Höchster and Huneke.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 42 , Issue 3 , October 1999 , pp. 541 - 549
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Bruns, W. and Herzog, J., Cohen-Macaulay rings (Cambridge University Press, 1993).Google Scholar
2.Hochster, M. and Huneke, C., Tight closure and strong F-regularity, Soc. Math. France Memoire 38 (1989), 119–133.CrossRefGoogle Scholar
3.Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31–116.Google Scholar
4.Hochster, M. and Huneke, C., F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1–62.Google Scholar
5.Lipman, J., Rational singularities with applications to algebraic surfaces and unique factorization, I.H.E.S. Publ. Math. 36 (1969), 195–279.CrossRefGoogle Scholar
6.MacCrimmon, B., Weak F-regularity is strong F-regularity for rings with isolated non-ℚ-Gorenstein points, Trans. Amer. Math. Soc., to appear.Google Scholar
7.Smith, K. E., Tight closure of parameter ideals, Invent. Math. 115 (1994), 41–60.CrossRefGoogle Scholar
8.Smith, K. E., Test ideals in local rings, Trans. Amer. Math. Soc. 347 (1995), 3453–3472.CrossRefGoogle Scholar
9.Smith, K. E., F-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159–180.CrossRefGoogle Scholar