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

Trung, Ngô

doi:10.1090/S0002-9947-98-02198-9

The Castelnuovo regularity of the Rees algebra and the associated graded ring
HTML articles powered by AMS MathViewer

by Ngô Viêt Trung

Trans. Amer. Math. Soc. 350 (1998), 2813-2832

DOI: https://doi.org/10.1090/S0002-9947-98-02198-9

PDF | Request permission

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.

References

Ian M. Aberbach, Craig Huneke, and Ngô Việt Trung, Reduction numbers, Briançon-Skoda theorems and the depth of Rees rings, Compositio Math. 97 (1995), no. 3, 403–434. MR 1353282
Douglas L. Costa, Sequences of linear type, J. Algebra 94 (1985), no. 1, 256–263. MR 789548, DOI 10.1016/0021-8693(85)90211-X
David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
Shiro Goto, Blowing-up of Buchsbaum rings, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 140–162. MR 693633
S. Goto, Y. Nakamura, and K. Nishida, Cohen-Macaulay graded rings associated to ideals, Amer. J. Math. 118 (1996), 1196-1213.
J. Herzog, A. Simis, and W. V. Vasconcelos, Approximation complexes of blowing-up rings, J. Algebra 74 (1982), no. 2, 466–493. MR 647249, DOI 10.1016/0021-8693(82)90034-5
J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
L.T. Hoa, Reduction numbers of equimultiple ideals, J. Pure Appl. Algebra 109 (1996), 111–126.
Sam Huckaba, Reduction numbers for ideals of higher analytic spread, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 49–57. MR 886434, DOI 10.1017/S0305004100067037
Sam Huckaba, On complete $d$-sequences and the defining ideals of Rees algebras, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 3, 445–458. MR 1010369, DOI 10.1017/S0305004100068183
Craig Huneke, On the symmetric and Rees algebra of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), no. 2, 268–275. MR 563225, DOI 10.1016/0021-8693(80)90179-9
Craig Huneke, Symbolic powers of prime ideals and special graded algebras, Comm. Algebra 9 (1981), no. 4, 339–366. MR 605026, DOI 10.1080/00927878108822586
Craig Huneke, Powers of ideals generated by weak $d$-sequences, J. Algebra 68 (1981), no. 2, 471–509. MR 608547, DOI 10.1016/0021-8693(81)90276-3
Craig Huneke, The theory of $d$-sequences and powers of ideals, Adv. in Math. 46 (1982), no. 3, 249–279. MR 683201, DOI 10.1016/0001-8708(82)90045-7
Mark Johnson and Bernd Ulrich, Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math. 103 (1996), no. 1, 7–29. MR 1404996
Bernard Johnston and Daniel Katz, Castelnuovo regularity and graded rings associated to an ideal, Proc. Amer. Math. Soc. 123 (1995), no. 3, 727–734. MR 1231300, DOI 10.1090/S0002-9939-1995-1231300-1
M. Kühl, Thesis, University of Essen, 1981.
Thomas Marley, The reduction number of an ideal and the local cohomology of the associated graded ring, Proc. Amer. Math. Soc. 117 (1993), no. 2, 335–341. MR 1112496, DOI 10.1090/S0002-9939-1993-1112496-7
Akira Ooishi, Castelnuovo’s regularity of graded rings and modules, Hiroshima Math. J. 12 (1982), no. 3, 627–644. MR 676563
Akira Ooishi, Genera and arithmetic genera of commutative rings, Hiroshima Math. J. 17 (1987), no. 1, 47–66. MR 886981
F. Planas-Vilanova, On the module of effective relations of a standard algebra, preprint.
K. Raghavan, Powers of ideals generated by quadratic sequences, Trans. Amer. Math. Soc. 343 (1994), no. 2, 727–747. MR 1188639, DOI 10.1090/S0002-9947-1994-1188639-1
Peter Schenzel, Castelnuovo’s index of regularity and reduction numbers, Topics in algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 201–208. MR 1171270
Ngô Việt Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), no. 2, 229–236. MR 902533, DOI 10.1090/S0002-9939-1987-0902533-1
Ngô Viêt Trung, Reduction number, $a$-invariant and Rees algebras of ideals having small analytic deviation, Commutative algebra (Trieste, 1992) World Sci. Publ., River Edge, NJ, 1994, pp. 245–262. MR 1421090
Dorin Popescu, Flatness in non-Noetherian ring theory, Rev. Roumaine Math. Pures Appl. 34 (1989), no. 9, 839–854. MR 1031703
Paolo Valabrega and Giuseppe Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93–101. MR 514892
Giuseppe Valla, On the symmetric and Rees algebras of an ideal, Manuscripta Math. 30 (1980), no. 3, 239–255. MR 557107, DOI 10.1007/BF01303330
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876