## On the arithmetic of tight closure

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- by Holger Brenner and Mordechai Katzman PDF
- J. Amer. Math. Soc.
**19**(2006), 659-672 Request permission

## 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

- Holger Brenner,
*Tight closure and projective bundles*, J. Algebra**265**(2003), no.Â 1, 45â78. MR**1984899**, DOI 10.1016/S0021-8693(03)00222-9 - Holger Brenner,
*A characteristic zero Hilbert-Kunz criterion for solid closure in dimension two*, Math. Res. Lett.**11**(2004), no.Â 5-6, 563â574. MR**2106225**, DOI 10.4310/MRL.2004.v11.n5.a1 - H. Brenner. The Hilbert-Kunz function in graded dimension two.
*ArXiv*, 2004. - Holger Brenner,
*Slopes of vector bundles on projective curves and applications to tight closure problems*, Trans. Amer. Math. Soc.**356**(2004), no.Â 1, 371â392. MR**2020037**, DOI 10.1090/S0002-9947-03-03391-9 - H. Brenner. On a problem of Miyaoka. In
*Number Fields and Function FieldsâTwo Parallel Worlds*, Progress in Math. 239, BirkhĂ€user, 51â59 (2005). - CoCoATeam. CoCoA, a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it.
- David Gieseker,
*$p$-ample bundles and their Chern classes*, Nagoya Math. J.**43**(1971), 91â116. MR**296078**, DOI 10.1017/S0027763000014380 - 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/.
- C. Han and P. Monsky,
*Some surprising Hilbert-Kunz functions*, Math. Z.**214**(1993), no.Â 1, 119â135. MR**1234602**, DOI 10.1007/BF02572395 - Nobuo Hara,
*A characterization of rational singularities in terms of injectivity of Frobenius maps*, Amer. J. Math.**120**(1998), no.Â 5, 981â996. MR**1646049**, DOI 10.1353/ajm.1998.0037 - Nobuo Hara,
*Geometric interpretation of tight closure and test ideals*, Trans. Amer. Math. Soc.**353**(2001), no.Â 5, 1885â1906. MR**1813597**, DOI 10.1090/S0002-9947-01-02695-2 - Robin Hartshorne,
*Ample vector bundles*, Inst. Hautes Ătudes Sci. Publ. Math.**29**(1966), 63â94. MR**193092** - Robin Hartshorne and Robert Speiser,
*Local cohomological dimension in characteristic $p$*, Ann. of Math. (2)**105**(1977), no.Â 1, 45â79. MR**441962**, DOI 10.2307/1971025 - Melvin Hochster,
*Solid closure*, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp.Â 103â172. MR**1266182**, DOI 10.1090/conm/159/01508 - Melvin Hochster,
*Tight closure in equal characteristic, big Cohen-Macaulay algebras, and solid closure*, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp.Â 173â196. MR**1266183**, DOI 10.1090/conm/159/01507 - M. Hochster and C. Huneke. Tight closure in equal zero characteristic. Preprint.
- Melvin Hochster and Craig Huneke,
*Tight closure, invariant theory, and the BrianĂ§on-Skoda theorem*, J. Amer. Math. Soc.**3**(1990), no.Â 1, 31â116. MR**1017784**, DOI 10.1090/S0894-0347-1990-1017784-6 - Craig Huneke,
*Tight closure and its applications*, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With an appendix by Melvin Hochster. MR**1377268**, DOI 10.1016/0167-4889(95)00136-0 - Craig Huneke,
*Tight closure, parameter ideals, and geometry*, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, BirkhĂ€user, Basel, 1998, pp.Â 187â239. MR**1648666** - Mordechai Katzman,
*Finite criteria for weak $F$-regularity*, Illinois J. Math.**40**(1996), no.Â 3, 453â463. MR**1407629** - Yoichi Miyaoka,
*The Chern classes and Kodaira dimension of a minimal variety*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp.Â 449â476. MR**946247**, DOI 10.2969/aspm/01010449 - T. Muir.
*The Theory of Determinants in the Historical Order of Development*, volume III. Macmillan, London, 1920. - Hans Schoutens,
*Non-standard tight closure for affine $\Bbb C$-algebras*, Manuscripta Math.**111**(2003), no.Â 3, 379â412. MR**1993501**, DOI 10.1007/s00229-003-0380-6 - Jean-Pierre Serre,
*Cours dâarithmĂ©tique*, Collection SUP: âLe MathĂ©maticienâ, vol. 2, Presses Universitaires de France, Paris, 1970 (French). MR**0255476** - N. I. Shepherd-Barron,
*Semi-stability and reduction mod $p$*, Topology**37**(1998), no.Â 3, 659â664. MR**1604907**, DOI 10.1016/S0040-9383(97)00038-4 - A. Singh and U. Walther. On the arithmetic rank of certain Segre products.
*ArXiv*, 2004. - Karen E. Smith,
*The multiplier ideal is a universal test ideal*, Comm. Algebra**28**(2000), no.Â 12, 5915â5929. Special issue in honor of Robin Hartshorne. MR**1808611**, DOI 10.1080/00927870008827196 - Karen E. Smith,
*Tight closure and vanishing theorems*, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000) ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp.Â 149â213. MR**1919458** - V. Trivedi. Hilbert-Kunz multiplicity and reduction mod $p$.
*ArXiv*, 2004. - V. van Zeipel. Om determinanter, hvars elementer Ă€ro binomialkoefficienter.
*Lunds Universitet*, Ă rsskrift ii:1â68, 1865.

## Additional Information

**Holger Brenner**- Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- MR Author ID: 322383
- 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
- Received by editor(s): December 3, 2004
- Published electronically: December 22, 2005
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2220102