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On the arithmetic of tight closure


Authors: Holger Brenner and Mordechai Katzman
Journal: J. Amer. Math. Soc. 19 (2006), 659-672
MSC (2000): Primary 13A35; Secondary 11A41, 14H60
DOI: https://doi.org/10.1090/S0894-0347-05-00514-X
Published electronically: December 22, 2005
MathSciNet review: 2220102
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Abstract: We provide a negative answer to an old question in tight closure theory by showing that the containment $ x^3y^3 \in (x^4,y^4,z^4)^*$ in $ \mathbb{K}[x,y,z]/(x^7+y^7-z^7)$ holds for infinitely many but not for almost all prime characteristics of the field $ \mathbb{K}$. This proves that tight closure exhibits a strong dependence on the arithmetic of the prime characteristic. The ideal $ (x,y,z) \subset \mathbb{K}[x,y,z,u,v,w]/(x^7+y^7-z^7, ux^4+vy^4+wz^4+x^3y^3)$ has then the property that the cohomological dimension fluctuates arithmetically between 0 and $ 1$.


References [Enhancements On Off] (What's this?)

  • 1. H. Brenner.
    Tight closure and projective bundles.
    J. Algebra, 265:45-78, 2003. MR 1984899 (2004h:13008)
  • 2. H. Brenner.
    A characteristic zero Hilbert-Kunz criterion for solid closure in dimension two.
    Math. Research Letters, 11:563-574, 2004. MR 2106225 (2005k:13008)
  • 3. H. Brenner.
    The Hilbert-Kunz function in graded dimension two.
    ArXiv, 2004.
  • 4. H. Brenner.
    Slopes of vector bundles and applications to tight closure problems.
    Trans. Amer. Math. Soc., 356(1):371-392, 2004. MR 2020037 (2004m:13017)
  • 5. H. Brenner.
    On a problem of Miyaoka. In Number Fields and Function Fields--Two Parallel Worlds, Progress in Math. 239, Birkhäuser, 51-59 (2005).
  • 6. CoCoATeam.
    CoCoA: a system for doing Computations in Commutative Algebra.
    Available at http://cocoa.dima.unige.it.
  • 7. D. Gieseker.
    $ p$-ample bundles and their Chern classes.
    Nagoya Math. J., 43:91-116, 1971. MR 0296078 (45:5139)
  • 8. D. R. Grayson and M. E. Stillman.
    Macaulay 2, a software system for research in algebraic geometry.
    Available at http://www.math.uiuc.edu/Macaulay2/.
  • 9. C. Han and P. Monsky.
    Some surprising Hilbert-Kunz functions.
    Math. Z., 214:119-135, 1993. MR 1234602 (94f:13008)
  • 10. N. Hara.
    A characterization of rational singularities in terms of injectivity of Frobenius maps.
    Amer. J. of Math., 120(5):981-996, 1998. MR 1646049 (99h:13005)
  • 11. N. Hara.
    Geometric interpretation of tight closure and test ideals.
    Trans. Amer. Math. Soc., 353(5):1885-1906, 2001. MR 1813597 (2001m:13009)
  • 12. R. Hartshorne.
    Ample vector bundles.
    Publ. Math. I.H.E.S., 29:63-94, 1966. MR 0193092 (33:1313)
  • 13. R. Hartshorne and R. Speiser.
    Local cohomological dimension in characteristic p.
    Ann. of Math., 105:45-79, 1977. MR 0441962 (56:353)
  • 14. M. Hochster.
    Solid closure.
    Contemp. Math., 159:103-172, 1994. MR 1266182 (95a:13011)
  • 15. M. Hochster.
    Tight closure in equal characteristic, big Cohen-Macaulay algebras, and solid closure.
    Contemp. Math., 159:173-196, 1994. MR 1266183 (95a:13012)
  • 16. M. Hochster and C. Huneke.
    Tight closure in equal zero characteristic. Preprint.
  • 17. M. Hochster and C. Huneke.
    Tight closure, invariant theory, and the Briançon-Skoda theorem.
    J. Amer. Math. Soc., 3:31-116, 1990. MR 1017784 (91g:13010)
  • 18. C. Huneke.
    Tight Closure and Its Applications, volume 88 of CBMS Lecture Notes in Mathematics.
    AMS, Providence, 1996. MR 1377268 (96m:13001)
  • 19. C. Huneke.
    Tight closure, parameter ideals, and geometry.
    In Six Lectures on Commutative Algebra. Birkhäuser, 1998. MR 1648666 (99j:13001)
  • 20. M. Katzman.
    Finite criteria for weak F-regularity.
    Illinois J. Math., 40(3):453-463, 1996. MR 1407629 (97m:13005)
  • 21. Y. Miyaoka.
    The Chern class and Kodaira dimension of a minimal variety.
    In Algebraic Geometry, Sendai 1985, volume 10 of Adv. Stud. Pure Math., pages 449-476, 1987. MR 0946247 (89k:14022)
  • 22. T. Muir.
    The Theory of Determinants in the Historical Order of Development, volume III.
    Macmillan, London, 1920.
  • 23. H. Schoutens.
    Non-standard tight closure for affine $ {\mathbb{C}}$-algebras.
    Manus. Math., 111:379-412, 2003. MR 1993501 (2004m:13019)
  • 24. J. P. Serre.
    Cours d' Arithmétique.
    Presses Universitaires de France, 1970. MR 0255476 (41:138)
  • 25. N. I. Shepherd-Barron.
    Semi-stability and reduction mod $ p$.
    Topology, 37(3):659-664, 1997. MR 1604907 (99c:14057)
  • 26. A. Singh and U. Walther.
    On the arithmetic rank of certain Segre products.
    ArXiv, 2004.
  • 27. K. E. Smith.
    The multiplier ideal is a universal test ideal.
    Commun. in Algebra, 28(12):5915-5929, 2000. MR 1808611 (2002d:13008)
  • 28. K. E. Smith.
    Tight closure and vanishing theorems.
    In Demailly, editor, School on vanishing theorems, volume 6 of ICTP Lect. Notes, pages 151-213, 2000. MR 1919458 (2003f:13005)
  • 29. V. Trivedi.
    Hilbert-Kunz multiplicity and reduction mod $ p$.
    ArXiv, 2004.
  • 30. V. van Zeipel.
    Om determinanter, hvars elementer äro binomialkoefficienter.
    Lunds Universitet, Årsskrift ii:1-68, 1865.

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Additional Information

Holger Brenner
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: H.Brenner@sheffield.ac.uk

Mordechai Katzman
Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
Email: M.Katzman@sheffield.ac.uk

DOI: https://doi.org/10.1090/S0894-0347-05-00514-X
Keywords: Tight closure, dependence on prime numbers, cohomological dimension, semistable bundles.
Received by editor(s): December 3, 2004
Published electronically: December 22, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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