Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Tight closure, plus closure
and Frobenius closure in cubical cones


Author: Moira A. McDermott
Journal: Trans. Amer. Math. Soc. 352 (2000), 95-114
MSC (1991): Primary 13A35, 13A02, 13H10
DOI: https://doi.org/10.1090/S0002-9947-99-02396-X
Published electronically: March 8, 1999
MathSciNet review: 1624198
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider tight closure, plus closure and Frobenius closure in the rings $R = K[[x,y,z]]/(x^{3} + y^{3} +z^{3})$, where $K$ is a field of characteristic $p$ and $p \neq 3$. We use a $\mathbb{Z}_3$-grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring $K[[x,y]]$. We show that Frobenius closure is the same as tight closure in certain classes of ideals when $p \equiv 2\ \text{mod}\ 3$. Since $I^{F} \subseteq IR^{+} \cap R \subseteq I^{*}$, we conclude that $IR^{+} \cap R = I^{*}$ for these ideals. Using injective modules over the ring $R^{\infty }$, the union of all ${p^{e}}$th roots of elements of $R$, we reduce the question of whether $I^{F} = I^{*}$ for $\mathbb{Z}_3 $-graded ideals to the case of $\mathbb{Z}_3$-graded irreducible modules. We classify the irreducible $m$-primary $\mathbb{Z}_3$-graded ideals. We then show that $I^{F} = I^{*}$ for most irreducible $m$-primary $\mathbb{Z}_3$-graded ideals in $K[[x,y,z]]/(x^3+y^3+z^3)$, where $K$ is a field of characteristic $p$ and $p \equiv 2\ \text{mod}\ 3$. Hence $I^{*} = IR^{+} \cap R$ for these ideals.


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

  • [Ab] I. Aberbach, Tight closure in F-rational rings, Nagoya Math. J. 135 (1994), 43-54. MR 95g:13020
  • [E] D. Eisenbud, Commutative Algebra With a View Toward Algebraic Geometry, Graduate Text in Mathematics 150, Springer-Verlag, New York, 1995. MR 97a:13001
  • [Fi] N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly, 54 (1947), 589-592. MR 9:331b
  • [HH1] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1) (1990), 31-116. MR 91g:13010
  • [HH2] M. Hochster and C. Huneke, Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geometry 3 (1994), 599-670. MR 95k:13002
  • [HR] M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117-172. MR 54:5230
  • [Hu] C. Huneke, Tight closure and its applications, C.B.M.S. Regional Conf. Ser. in Math. No. 88 (1996). MR 96m:13001
  • [HuS] C. Huneke and K. E. Smith, Tight closure and the Kodaira vanishing theorem, J. Reine Angew. Math. 484 (1997), 127-152. MR 98e:13007
  • [K] E. Kunz, Characterizations of regular local rings of characteristic p, Amer. J. Math. 91 (1969), 772-784. MR 40:5609
  • [L] E. Lucas, Théorie des functions numérique simplement pèriodiques, Amer. J. Math. 1 (1878), 184-240.
  • [N] M. Nagata, Local Rings, Interscience, New York, 1962. MR 27:5790
  • [Si] A. Singh, A computation of tight closure in diagonal hypersurfaces, Journal of Algebra 203 (1998), 579-589. CMP 98:12
  • [Sm1] K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1) (1994), 41-60. MR 94k:13006
  • [Sm2] K. E. Smith, Test ideals in local rings, Trans. Amer. Math. Soc. 347 (9) (1995), 3453-3472. MR 96c:13008
  • [Sm3] K. E. Smith, Tight closure in graded rings, J. Math. Kyoto Univ. 37 (1) (1997), 35-53. MR 98e:13009

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13A35, 13A02, 13H10

Retrieve articles in all journals with MSC (1991): 13A35, 13A02, 13H10


Additional Information

Moira A. McDermott
Affiliation: Mathematics and Computer Science Department, Gustavus Adolphus College, 800 W. College Avenue, St. Peter, Minnesota 56082-1498
Email: mmcdermo@gac.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02396-X
Keywords: Tight closure, characteristic $p$, Frobenius closure, plus closure
Received by editor(s): August 27, 1997
Published electronically: March 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society