Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules

Authors:
M. Brodmann and F. Rohrer

Journal:
Proc. Amer. Math. Soc. **133** (2005), 987-993

MSC (2000):
Primary 13D45, 13E10

DOI:
https://doi.org/10.1090/S0002-9939-04-07779-2

Published electronically:
November 19, 2004

MathSciNet review:
2117198

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Noetherian homogeneous ring with one-dimensional local base ring . Let be an -primary ideal, let be a finitely generated graded -module and let . Let denote the -th local cohomology module of with respect to the irrelevant ideal of . We show that the first Hilbert-Samuel coefficient of the -th graded component of with respect to is antipolynomial of degree in . In addition, we prove that the postulation numbers of the components with respect to have a common upper bound.

**[B-F-L]**M. Brodmann, S. Fumasoli, and C. S. Lim,*Low-codimensional associated primes of graded components of local cohomology modules*, J. Algebra**275**(2004), no. 2, 867–882. MR**2052643**, https://doi.org/10.1016/j.jalgebra.2003.12.003**[B-F-T]**M. Brodmann, S. Fumasoli, and R. Tajarod,*Local cohomology over homogeneous rings with one-dimensional local base ring*, Proc. Amer. Math. Soc.**131**(2003), no. 10, 2977–2985. MR**1993202**, https://doi.org/10.1090/S0002-9939-03-07009-6**[B-H]**M. Brodmann and M. Hellus,*Cohomological patterns of coherent sheaves over projective schemes*, J. Pure Appl. Algebra**172**(2002), no. 2-3, 165–182. MR**1906872**, https://doi.org/10.1016/S0022-4049(01)00144-X**[B-K-S]**Markus P. Brodmann, Mordechai Katzman, and Rodney Y. Sharp,*Associated primes of graded components of local cohomology modules*, Trans. Amer. Math. Soc.**354**(2002), no. 11, 4261–4283. MR**1926875**, https://doi.org/10.1090/S0002-9947-02-02987-2**[B-S]**M. P. Brodmann and R. Y. Sharp,*Local cohomology: an algebraic introduction with geometric applications*, Cambridge Studies in Advanced Mathematics, vol. 60, Cambridge University Press, Cambridge, 1998. MR**1613627****[E]**David Eisenbud,*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960****[K]**Mordechai Katzman,*An example of an infinite set of associated primes of a local cohomology module*, J. Algebra**252**(2002), no. 1, 161–166. MR**1922391**, https://doi.org/10.1016/S0021-8693(02)00032-7**[Ki]**D. Kirby,*Artinian modules and Hilbert polynomials*, Quart. J. Math. Oxford Ser. (2)**24**(1973), 47–57. MR**0316446**, https://doi.org/10.1093/qmath/24.1.47**[L]**C.S. LIM:*Graded local cohomology modules and their associated primes*, Communications in Algebra 32, No 2 (2004), 727-745.**[S]**Anurag K. Singh,*𝑝-torsion elements in local cohomology modules*, Math. Res. Lett.**7**(2000), no. 2-3, 165–176. MR**1764314**, https://doi.org/10.4310/MRL.2000.v7.n2.a3**[T]**N.V. TRUNG:*Reduction exponent and degree bound for the defining equations of graded rings*, Proceedings of the AMS 101 (1987), 229 -236. MR**0902533 (89i:13031)**

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Additional Information

**M. Brodmann**

Affiliation:
Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Email:
brodmann@math.unizh.ch

**F. Rohrer**

Affiliation:
Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Email:
fred@math.unizh.ch

DOI:
https://doi.org/10.1090/S0002-9939-04-07779-2

Keywords:
Local cohomology modules,
graded components,
Hilbert-Samuel polynomials

Received by editor(s):
December 1, 2003

Published electronically:
November 19, 2004

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2004
American Mathematical Society