A theorem of Briançon-Skoda type for regular local rings containing a field
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- by Ian M. Aberbach and Craig Huneke PDF
- Proc. Amer. Math. Soc. 124 (1996), 707-713 Request permission
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
- Ian M. Aberbach and Craig Huneke, An improved Briançon-Skoda theorem with applications to the Cohen-Macaulayness of Rees algebras, Math. Ann. 297 (1993), no. 2, 343–369. MR 1241812, DOI 10.1007/BF01459507
- 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.
- M. Artin, Algebraic structure of power series rings, Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981) Contemp. Math., vol. 13, Amer. Math. Soc., Providence, R.I., 1982, pp. 223–227. MR 685955
- Henri Skoda and Joël Briançon, Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de $\textbf {C}^{n}$, C. R. Acad. Sci. Paris Sér. A 278 (1974), 949–951 (French). MR 340642
- Melvin Hochster and Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784, DOI 10.1090/S0894-0347-1990-1017784-6
- Craig Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), no. 2, 293–318. MR 894879, DOI 10.1307/mmj/1029003560
- C. Huneke and I. Swanson, Cores of ideals in two-dimensional regular local rings, Michigan Math. J. 42 (1995), 193–208.
- Shiroh Itoh, Integral closures of ideals generated by regular sequences, J. Algebra 117 (1988), no. 2, 390–401. MR 957448, DOI 10.1016/0021-8693(88)90114-7
- J. Lipman, Adjoints of ideals in regular local rings, Math. Res. Letters 1 (1994), 1–17.
- Joseph Lipman and Avinash Sathaye, Jacobian ideals and a theorem of Briançon-Skoda, Michigan Math. J. 28 (1981), no. 2, 199–222. MR 616270
- Joseph Lipman and Bernard Teissier, Pseudorational local rings and a theorem of Briançon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981), no. 1, 97–116. MR 600418
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- D. Rees and Judith D. Sally, General elements and joint reductions, Michigan Math. J. 35 (1988), no. 2, 241–254. MR 959271, DOI 10.1307/mmj/1029003751
- M. Spivakovsky, Smoothing of ring homomorphisms, approximation theorems and the Bass-Quillen conjecture, preprint.
- Irena Swanson, Joint reductions, tight closure, and the Briançon-Skoda theorem, J. Algebra 147 (1992), no. 1, 128–136. MR 1154678, DOI 10.1016/0021-8693(92)90256-L
Additional Information
- Ian M. Aberbach
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 314830
- Email: aberbach@msindy8.cs.missouri.edu
- Craig Huneke
- Affiliation: Department of Mathematics, Purdue University, W. Lafayette, Indiana 47907
- MR Author ID: 89875
- Email: huneke@math.purdue.edu
- 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 707-713
- MSC (1991): Primary 13H05; Secondary 13A35, 13B22
- DOI: https://doi.org/10.1090/S0002-9939-96-03058-4
- MathSciNet review: 1301483