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The Castelnuovo regularity of the Rees algebra
and the associated graded ring


Author: Ngô Viêt Trung
Journal: Trans. Amer. Math. Soc. 350 (1998), 2813-2832
MSC (1991): Primary 13A30; Secondary 13D45
DOI: https://doi.org/10.1090/S0002-9947-98-02198-9
MathSciNet review: 1473456
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Abstract: It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of $d$-sequences.


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  • 1. I. M. Aberbach, C. Huneke, and Ngô Viêt Trung, Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings, Compositio Math. 97 (1995), 403-434. MR 96g:13002
  • 2. D.G. Costa, Sequences of linear type, J. Algebra 94 (1985), 256-263. MR 86h:13010
  • 3. D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89-133. MR 85f:13023
  • 4. S. Goto, Blowing-up of Buchsbaum rings, In: Commutative Algebra, London Math. Soc. Lecture Notes 72, Cambridge Univ. Press, 1982, 140-162. MR 84h:13032
  • 5. S. Goto, Y. Nakamura, and K. Nishida, Cohen-Macaulay graded rings associated to ideals, Amer. J. Math. 118 (1996), 1196-1213. CMP 97:04
  • 6. J. Herzog, A. Simis, and W. Vasconcelos, Approximation complexes of blowing-up rings I, J. Algebra 18 (1982), 466-493. MR 83h:13023
  • 7. J. Herzog, A. Simis, and W. Vasconcelos, Koszul cohomology and blowing-up rings, in: Commutative Algebra, Marcel Dekker, New York, 1983, 79-169. MR 84k:13015
  • 8. L.T. Hoa, Reduction numbers of equimultiple ideals, J. Pure Appl. Algebra 109 (1996), 111-126.
  • 9. S. Huckaba, Reduction numbers for ideals of higher analytic spread, Math. Proc. Camb. Phil. Soc. 102 (1987), 49-57. MR 89b:13023
  • 10. S. Huckaba, On complete $d$-sequences and the defining ideals of Rees algebras, Math. Proc. Camb. Phil. Soc. 106 (1989), 445-458. MR 90m:13012
  • 11. C. Huneke, On the symmetric and Rees algebras of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), 268-275. MR 81d:13016
  • 12. C. Huneke, Symbolic powers of prime ideals and special graded algebras, Commun. Algebra 9 (1981), 339-366. MR 83a:13011
  • 13. C. Huneke, Powers of ideals generated by weak $d$-sequences, J. Algebra 62 (1981), 471-509. MR 82k:13003
  • 14. C. Huneke, The theory of $d$-sequences and powers of ideals, Adv. in Math. 46 (1982), 249-279. MR 84g:13021
  • 15. M. Johnson and B. Ulrich, Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math. 108 (1996), 7-29. MR 97f:13006
  • 16. B. Johnston and D. Katz, Castelnuovo regularity and graded rings associated to an ideal, Proc. Amer. Math. Soc. 123 (1995), 727-734. MR 95d:13005
  • 17. M. Kühl, Thesis, University of Essen, 1981.
  • 18. T. Marley, The reduction number of an ideal and the local cohomology of the associated graded ring, Proc. Amer. Math. Soc. 117 (1993), 335-341. MR 93d:13029
  • 19. A. Ooishi, Castelnuovo's regularity of graded rings and modules, Hiroshima Math. J. 12 (1982), 627-644. MR 84m:13024
  • 20. A. Ooishi, Genera and arithmetic genera of commutative rings, Hiroshima Math. J. 17 (1987), 47-66. MR 89a:13025
  • 21. F. Planas-Vilanova, On the module of effective relations of a standard algebra, preprint.
  • 22. K.N. Raghavan, Powers of ideals generated by quadratic sequences, Trans. Amer. Math. Soc. 343 (1994), 727-747. MR 94i:13007
  • 23. P. Schenzel, Castelnuovo's index of regularity and reduction numbers, In: Topics in Algebra, Part II, Banach Center Publication 26, Warsaw, 1990, 201-208. MR 93g:13013
  • 24. Ngô Viêt Trung, Reduction exponent and bounds for the degree of the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), 229-235. MR 89i:13031
  • 25. Ngô Viêt Trung, Reduction numbers, $a$-invariants, and Rees algebras of ideals having small analytic deviation, In: Commutative Algebra, World Scientific, 1994, 245-262. MR 97f:13007
  • 26. Ngô Viêt Trung and S. Ikeda, When is the Rees algebra Cohen-Macaulay? Comm. Algebra 17 (1989), 2893-2922. MR 91a:13008
  • 27. P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93-101. MR 80d:14010
  • 28. G. Valla, On the symmetric and Rees algebras of an ideal, Manuscripta Math. 30 (1980), 239-255; 33 (1981), 59-61. MR 83b:14017; MR 83b:14018
  • 29. O. Zariski and P. Samuel, Commutative Algebra II, Springer, 1975. MR 52:10706

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Additional Information

Ngô Viêt Trung
Affiliation: Institute of Mathematics, Box 631, Bò Hô, Hanoi, Vietnam
Email: nvtrung@thevinh.ac.vn

DOI: https://doi.org/10.1090/S0002-9947-98-02198-9
Keywords: Rees algebra, associated graded ring, Castelnuovo regularity, relation type, reduction number, $d$-sequence, filter-regular sequence
Received by editor(s): June 15, 1996
Additional Notes: This work is partially supported by the National Basic Research Program of Vietnam
Dedicated: Dedicated to the memory of Professor Hideyuki Matsumura
Article copyright: © Copyright 1998 American Mathematical Society

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