Coefficient ideals and the Cohen-Macaulay property of Rees algebras
Author:
Eero Hyry
Journal:
Proc. Amer. Math. Soc. 129 (2001), 1299-1308
MSC (2000):
Primary 13A30; Secondary 13B22, 14B05
DOI:
https://doi.org/10.1090/S0002-9939-00-05673-2
Published electronically:
October 24, 2000
MathSciNet review:
1712905
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Let be a local ring and let
be an ideal of positive height. If
is a reduction of
, then the coefficient ideal
is by definition the largest ideal
such that
. In this article we study the ideal
when the Rees algebra
is Cohen-Macaulay.
- 1. Ian M. Aberbach and Craig Huneke, A theorem of Briançon-Skoda type for regular local rings containing a field, Proc. Amer. Math. Soc. 124 (1996), no. 3, 707–713. MR 1301483, https://doi.org/10.1090/S0002-9939-96-03058-4
- 2. 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
- 3. Shiro Goto and Koji Nishida, The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations, American Mathematical Society, Providence, RI, 1994. Mem. Amer. Math. Soc. 110 (1994), no. 526. MR 1287443
- 4. Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. MR 0222093
- 5. M. Herrmann, E. Hyry, and T. Korb, On Rees algebras with a Gorenstein Veronese subring, J. Algebra 200 (1998), no. 1, 279–311. MR 1603275, https://doi.org/10.1006/jabr.1997.7207
- 6. M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag, Berlin, 1988. An algebraic study; With an appendix by B. Moonen. MR 954831
- 7. J. Herzog, A. Simis, and W. V. Vasconcelos, On the canonical module of the Rees algebra and the associated graded ring of an ideal, J. Algebra 105 (1987), no. 2, 285–302. MR 873664, https://doi.org/10.1016/0021-8693(87)90194-3
- 8. 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, https://doi.org/10.1090/S0894-0347-1990-1017784-6
- 9. E. Hyry, Blow-up algebras and rational singularities, Manus. Math. 98 (1999), 377-390. CMP 2000:03
- 10. -, Necessary and sufficient conditions for the Cohen-Macaulayness of the form ring, J. Algebra 212 (1999), 17-27. CMP 99:07
- 11. Manfred Herrmann, Eero Hyry, and Jürgen Ribbe, On multi-Rees algebras, Math. Ann. 301 (1995), no. 2, 249–279. With an appendix by Ngô Viêt Trung. MR 1314587, https://doi.org/10.1007/BF01446629
- 12. Mark Johnson and Bernd Ulrich, Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math. 103 (1996), no. 1, 7–29. MR 1404996
- 13. Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
- 14. Joseph Lipman, Desingularization of two-dimensional schemes, Ann. Math. (2) 107 (1978), no. 1, 151–207. MR 0491722, https://doi.org/10.2307/1971141
- 15. Joseph Lipman, Adjoints of ideals in regular local rings, Math. Res. Lett. 1 (1994), no. 6, 739–755. With an appendix by Steven Dale Cutkosky. MR 1306018, https://doi.org/10.4310/MRL.1994.v1.n6.a10
- 16. Joseph Lipman, Cohen-Macaulayness in graded algebras, Math. Res. Lett. 1 (1994), no. 2, 149–157. MR 1266753, https://doi.org/10.4310/MRL.1994.v1.n2.a2
- 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
- 18. 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
- 19. Aron Simis, Bernd Ulrich, and Wolmer V. Vasconcelos, Cohen-Macaulay Rees algebras and degrees of polynomial relations, Math. Ann. 301 (1995), no. 3, 421–444. MR 1324518, https://doi.org/10.1007/BF01446637
- 20. Ngô Vi\cfudot{e}t Trung, Duong Quôc Viêt, and Santiago Zarzuela, When is the Rees algebra Gorenstein?, J. Algebra 175 (1995), no. 1, 137–156. MR 1338971, https://doi.org/10.1006/jabr.1995.1179
- 21. Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR 1275840
- 22. -, Lecture notes on Cohen-Macaulay blowup algebras and their equations, Lecture notes for the Workshop on Commutative Algebra and its Relation to Combinatorics and Computer Algebra in Trieste, 1994.
- 23. Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition; Graduate Texts in Mathematics, Vol. 29. MR 0389876
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Additional Information
Eero Hyry
Affiliation:
Department of Technology, National Defense College, Santahamina, FIN-00860, Helsinki, Finland
Email:
Eero.Hyry@helsinki.fi
DOI:
https://doi.org/10.1090/S0002-9939-00-05673-2
Received by editor(s):
April 12, 1999
Received by editor(s) in revised form:
August 24, 1999
Published electronically:
October 24, 2000
Communicated by:
Wolmer V. Vasconcelos
Article copyright:
© Copyright 2000
American Mathematical Society