A theorem of BriançonSkoda type for regular local rings containing a field
Authors:
Ian M. Aberbach and Craig Huneke
Journal:
Proc. Amer. Math. Soc. 124 (1996), 707713
MSC (1991):
Primary 13H05; Secondary 13A35, 13B22
MathSciNet review:
1301483
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Abstract 
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Abstract: Let be a regular local ring containing a field. We give a refinement of the BriançonSkoda theorem showing that if is a minimal reduction of where is primary, then where and is the largest ideal such that . The proof uses tight closure in characteristic and reduction to characteristic for rings containing the rationals.
 [AH]
Ian
M. Aberbach and Craig
Huneke, An improved BriançonSkoda theorem with applications
to the CohenMacaulayness of Rees algebras, Math. Ann.
297 (1993), no. 2, 343–369. MR 1241812
(95b:13005), http://dx.doi.org/10.1007/BF01459507
 [AHT]
I. M. Aberbach, C. Huneke, and N. V. Trung, Reduction numbers, BriançonSkoda theorems and the depth of Rees rings, Compositio Math. 97 (1995), 403434.
 [Ar]
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
(84b:13014)
 [BS]
Henri
Skoda and Joël
Briançon, Sur la clôture intégrale d’un
idéal de germes de fonctions holomorphes en un point de
𝐶ⁿ, C. R. Acad. Sci. Paris Sér. A
278 (1974), 949–951 (French). MR 0340642
(49 #5394)
 [HH]
Melvin
Hochster and Craig
Huneke, Tight closure, invariant theory, and
the BriançonSkoda theorem, J. Amer.
Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784
(91g:13010), http://dx.doi.org/10.1090/S08940347199010177846
 [Hu]
Craig
Huneke, Hilbert functions and symbolic powers, Michigan Math.
J. 34 (1987), no. 2, 293–318. MR 894879
(89b:13037), http://dx.doi.org/10.1307/mmj/1029003560
 [HS]
C. Huneke and I. Swanson, Cores of ideals in twodimensional regular local rings, Michigan Math. J. 42 (1995), 193208.
 [It]
Shiroh
Itoh, Integral closures of ideals generated by regular
sequences, J. Algebra 117 (1988), no. 2,
390–401. MR
957448 (90g:13013), http://dx.doi.org/10.1016/00218693(88)901147
 [L]
J. Lipman, Adjoints of ideals in regular local rings, Math. Res. Letters 1 (1994), 117. CMP 95:05
 [LS]
Joseph
Lipman and Avinash
Sathaye, Jacobian ideals and a theorem of
BriançonSkoda, Michigan Math. J. 28 (1981),
no. 2, 199–222. MR 616270
(83m:13001)
 [LT]
Joseph
Lipman and Bernard
Teissier, Pseudorational local rings and a theorem of
BriançonSkoda about integral closures of ideals, Michigan
Math. J. 28 (1981), no. 1, 97–116. MR 600418
(82f:14004)
 [NR]
D.
G. Northcott and D.
Rees, Reductions of ideals in local rings, Proc. Cambridge
Philos. Soc. 50 (1954), 145–158. MR 0059889
(15,596a)
 [RS]
D.
Rees and Judith
D. Sally, General elements and joint reductions, Michigan
Math. J. 35 (1988), no. 2, 241–254. MR 959271
(89h:13034), http://dx.doi.org/10.1307/mmj/1029003751
 [Sp]
M. Spivakovsky, Smoothing of ring homomorphisms, approximation theorems and the BassQuillen conjecture, preprint.
 [Sw]
Irena
Swanson, Joint reductions, tight closure, and the
BriançonSkoda theorem, J. Algebra 147 (1992),
no. 1, 128–136. MR 1154678
(93g:13001), http://dx.doi.org/10.1016/00218693(92)90256L
 [AH]
 I. M. Aberbach and C. Huneke, An improved BriançonSkoda theorem with applications to the CohenMacaulayness of Rees rings, Math. Ann. 297 (1993), 343369. MR 95b:13005
 [AHT]
 I. M. Aberbach, C. Huneke, and N. V. Trung, Reduction numbers, BriançonSkoda theorems and the depth of Rees rings, Compositio Math. 97 (1995), 403434.
 [Ar]
 M. Artin, Algebraic structure of power series rings, Contemp. Math. 13 (1982), 223227. 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. R. Acad. Sci. Paris Sér. A 278 (1974), 949951. MR 49:5394
 [HH]
 M. Hochster and C. Huneke, Tight closure, invariant theory, and the BriançonSkoda theorem, J. Amer. Math. Soc. 3 (1990), 31116. MR 91g:13010
 [Hu]
 C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293318. MR 89b:13037
 [HS]
 C. Huneke and I. Swanson, Cores of ideals in twodimensional regular local rings, Michigan Math. J. 42 (1995), 193208.
 [It]
 S. Itoh, Integral closures of ideals generated by regular sequences, J. Algebra 117 (1988), 390401. MR 90g:13013
 [L]
 J. Lipman, Adjoints of ideals in regular local rings, Math. Res. Letters 1 (1994), 117. CMP 95:05
 [LS]
 J. Lipman and A. Sathaye, Jacobian ideals and a theorem of BriançonSkoda, Michigan Math. J. 28 (1981). MR 83m:13001
 [LT]
 J. Lipman and B. Teissier, Pseudorational local rings and a theorem of BriançonSkoda about integral closures of ideals, Michigan Math. J. 28 (1981), 97116. MR 82f:14004
 [NR]
 S. Northcott and D. Rees, Reductions of ideals in local rings, Math. Proc. Cambridge Philos. Soc. 50 (1954), 145158. MR 15:596a
 [RS]
 D. Rees and J. Sally, General elements and joint reductions, Michigan Math. J. 35 (1988), 241254. MR 89h:13034
 [Sp]
 M. Spivakovsky, Smoothing of ring homomorphisms, approximation theorems and the BassQuillen conjecture, preprint.
 [Sw]
 I. Swanson, Joint reductions, tight closure, and the BriançonSkoda theorem, J. Algebra 147 (1992), 128136. MR 93g:13001
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Additional Information
Ian M. Aberbach
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email:
aberbach@msindy8.cs.missouri.edu
Craig Huneke
Affiliation:
Department of Mathematics, Purdue University, W. Lafayette, Indiana 47907
Email:
huneke@math.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0002993996030584
PII:
S 00029939(96)030584
Keywords:
BrianconSkoda 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
