Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A theorem of Briançon-Skoda type
for regular local rings containing a field

Authors: Ian M. Aberbach and Craig Huneke
Journal: Proc. Amer. Math. Soc. 124 (1996), 707-713
MSC (1991): Primary 13H05; Secondary 13A35, 13B22
MathSciNet review: 1301483
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $(R,m)$ be a regular local ring containing a field. We give a refinement of the Briançon-Skoda theorem showing that if $J$ is a minimal reduction of $I$ where $I$ is $m$-primary, then $\overline{I^{d+w}} \subseteq J^{w+1}\mathfrak{a}$ where $d = \dim R$ and $\mathfrak{a}$ is the largest ideal such that $\mathfrak{a} J = \mathfrak{a} I$. The proof uses tight closure in characteristic $p$ and reduction to characteristic $p$ for rings containing the rationals.

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

  • [AH] I. M. Aberbach and C. Huneke, An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees rings, Math. Ann. 297 (1993), 343--369. MR 95b:13005
  • [AHT] I. M. Aberbach, C. Huneke, and N. V. Trung, Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings, Compositio Math. 97 (1995), 403--434.
  • [Ar] M. Artin, Algebraic structure of power series rings, Contemp. Math. 13 (1982), 223--227. MR 84b:13014
  • [BS] J. Briançon and H. Skoda, Sur la clôture intégrale d'un idéal de germes de fonctions holomorphes en un point de $C^{n}$, C. R. Acad. Sci. Paris Sér. A 278 (1974), 949--951. MR 49:5394
  • [HH] M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31--116. MR 91g:13010
  • [Hu] C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293--318. MR 89b:13037
  • [HS] C. Huneke and I. Swanson, Cores of ideals in two-dimensional regular local rings, Michigan Math. J. 42 (1995), 193--208.
  • [It] S. Itoh, Integral closures of ideals generated by regular sequences, J. Algebra 117 (1988), 390--401. MR 90g:13013
  • [L] J. Lipman, Adjoints of ideals in regular local rings, Math. Res. Letters 1 (1994), 1--17. CMP 95:05
  • [LS] J. Lipman and A. Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981). MR 83m:13001
  • [LT] J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97--116. MR 82f:14004
  • [NR] S. Northcott and D. Rees, Reductions of ideals in local rings, Math. Proc. Cambridge Philos. Soc. 50 (1954), 145--158. MR 15:596a
  • [RS] D. Rees and J. Sally, General elements and joint reductions, Michigan Math. J. 35 (1988), 241--254. MR 89h:13034
  • [Sp] M. Spivakovsky, Smoothing of ring homomorphisms, approximation theorems and the Bass-Quillen conjecture, preprint.
  • [Sw] I. Swanson, Joint reductions, tight closure, and the Briançon-Skoda theorem, J. Algebra 147 (1992), 128--136. MR 93g:13001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13H05, 13A35, 13B22

Retrieve articles in all journals with MSC (1991): 13H05, 13A35, 13B22

Additional Information

Ian M. Aberbach
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Craig Huneke
Affiliation: Department of Mathematics, Purdue University, W. Lafayette, Indiana 47907

Keywords: Briancon-Skoda theorems, integral closure, tight closure
Received by editor(s): June 21, 1994
Received by editor(s) in revised form: September 7, 1994
Additional Notes: Both authors were partially supported by the National Science Foundation.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1996 American Mathematical Society