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Gröbner bases, local cohomology and reduction number


Author: Ngô Viêt Trung
Journal: Proc. Amer. Math. Soc. 129 (2001), 9-18
MSC (1991): Primary 13P10; Secondary 13D45
DOI: https://doi.org/10.1090/S0002-9939-00-05503-9
Published electronically: June 21, 2000
MathSciNet review: 1695103
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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.


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  • [AH1] A. Aramova and J. Herzog, Koszul cycles and Eliahou-Kervaire type resolutions, J. Algebra 181 (1996), 347-370. MR 97c:13009
  • [AH2] A. Aramova and J. Herzog, Almost regular sequences and Betti numbers, preprint.
  • [BCP] D. Bayer, H. Charalambous and S. Popescu, Extremal Betti numbers and applications to monomial ideals, preprint.
  • [BS1] D. Bayer and M. Stillman, A theorem on refining divison orders by the reverse lexicographic orders, Duke J. Math. 55 (1987), 321-328. MR 87k:13005
  • [BS2] D. Bayer and M. Stillman, A criterion for detecting $m$-regularity, Invent. Math. 87 (1987), 1-11. MR 87k:13019
  • [BH] H. Bresinsky and L.T. Hoa, On the reduction number of some graded algebras, Proc. Amer. Math. Soc. 127 (1999), 1257-1263. MR 99h:13027
  • [E] D. Eisenbud, Commutative Algebra. With a view toward Algebraic Geometry, Springer, 1995. MR 97a:13001
  • [EG] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicities, J. Algebra 88 (1984), 89-133. MR 85f:13023
  • [EK] S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1-25. MR 91b:13019
  • [Ga] A. Galligo, Théorème de division et stabilité en géometrie analytique locale, Ann. Inst. Fourier 29 (1979), 107-184. MR 81e:32009
  • [GW] S. Goto and K. Watanabe, On graded rings, I, J. Math. Soc. Japan 30 (1978), 179-213. MR 81m:13021
  • [O] A. Ooishi, Castelnuovo regularity of graded rings and modules, Hiroshima Math. J. 12 (1982), 627-644. MR 84m:13024
  • [SCT] P. Schenzel, N. V. Trung, and N. T. Cuong, Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85 (1978), 57-73. MR 80i:13008
  • [T1] N. V. Trung, Reduction exponent and degree bounds for the defining equations of a graded ring, Proc. Amer. Math. Soc. 101 (1987), 229-236. MR 89i:13031
  • [T2] N. V. Trung, The Castelnuovo regularity of the Rees algebra and the associated graded ring, Trans. Amer. Math. Soc. 350 (1998), 2813-2832. MR 98j:13006
  • [T3] N. V. Trung, The largest non-vanishing degree of graded local cohomology modules, J. Algebra 215 (1999), 481-499. CMP 99:12
  • [V] W. Vasconcelos, Cohomological degrees of graded modules, in: Six Lectures on Commutative Algebra, Progress in Mathematics 166, Birkhäuser, Boston, 1998, 345-392. CMP 99:02

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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-00-05503-9
Keywords: Local cohomology, initial ideal, Borel-fixed ideal, reduction number
Received by editor(s): December 9, 1998
Received by editor(s) in revised form: March 11, 1999
Published electronically: June 21, 2000
Additional Notes: The author is partially supported by the National Basic Research Program
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society

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