Gröbner bases, local cohomology and reduction number

Trung, Ngô

doi:10.1090/S0002-9939-00-05503-9

Gröbner bases, local cohomology and reduction number
HTML articles powered by AMS MathViewer

by Ngô Viêt Trung

Proc. Amer. Math. Soc. 129 (2001), 9-18

DOI: https://doi.org/10.1090/S0002-9939-00-05503-9

Published electronically: June 21, 2000

PDF | Request permission

Abstract:

D. Bayer and M. Stillman showed that Gröbner bases can be used to compute the Castelnuovo-Mumford regularity which is a measure for the vanishing of graded local cohomology modules. The aim of this paper is to show that the same method can be applied to study other cohomological invariants as well as the reduction number.

References

Annetta Aramova and Jürgen Herzog, Koszul cycles and Eliahou-Kervaire type resolutions, J. Algebra 181 (1996), no. 2, 347–370. MR 1383471, DOI 10.1006/jabr.1996.0124
A. Aramova and J. Herzog, Almost regular sequences and Betti numbers, preprint.
D. Bayer, H. Charalambous and S. Popescu, Extremal Betti numbers and applications to monomial ideals, preprint.
S. Caenepeel, On Brauer groups of graded Krull domains and positively graded rings, J. Algebra 99 (1986), no. 2, 466–474. MR 837556, DOI 10.1016/0021-8693(86)90039-6
David Bayer and Michael Stillman, A criterion for detecting $m$-regularity, Invent. Math. 87 (1987), no. 1, 1–11. MR 862710, DOI 10.1007/BF01389151
Henrik Bresinsky and Lê Tuân Hoa, On the reduction number of some graded algebras, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1257–1263. MR 1473657, DOI 10.1090/S0002-9939-99-04622-5
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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
Shalom Eliahou and Michel Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), no. 1, 1–25. MR 1037391, DOI 10.1016/0021-8693(90)90237-I
André Galligo, Théorème de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 2, vii, 107–184 (French, with English summary). MR 539695
Shiro Goto and Keiichi Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179–213. MR 494707, DOI 10.2969/jmsj/03020179
Akira Ooishi, Castelnuovo’s regularity of graded rings and modules, Hiroshima Math. J. 12 (1982), no. 3, 627–644. MR 676563
Peter Schenzel, Ngô Viêt Trung, and Nguyễn Tụ’ Cu’ò’ng, Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85 (1978), 57–73 (German). MR 517641, DOI 10.1002/mana.19780850106
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, The Castelnuovo regularity of the Rees algebra and the associated graded ring, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2813–2832. MR 1473456, DOI 10.1090/S0002-9947-98-02198-9
N. V. Trung, The largest non-vanishing degree of graded local cohomology modules, J. Algebra 215 (1999), 481-499.
W. Vasconcelos, Cohomological degrees of graded modules, in: Six Lectures on Commutative Algebra, Progress in Mathematics 166, Birkhäuser, Boston, 1998, 345-392.