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

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- by M. Brodmann and F. Rohrer
- Proc. Amer. Math. Soc.
**133**(2005), 987-993 - DOI: https://doi.org/10.1090/S0002-9939-04-07779-2
- Published electronically: November 19, 2004
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## Abstract:

Let $R = \bigoplus _{n \geq 0} R_n$ be a Noetherian homogeneous ring with one-dimensional local base ring $(R_0, {\mathfrak m}_0)$. Let ${\mathfrak q}_0 \subseteq R_0$ be an ${\mathfrak m}_0$-primary ideal, let $M$ be a finitely generated graded $R$-module and let $i \in {\mathbb N}_0$. Let $H^i_{R_+}(M)$ denote the $i$-th local cohomology module of $M$ with respect to the irrelevant ideal $R_+:= \bigoplus _{n > 0} R_n$ of $R$. We show that the first Hilbert-Samuel coefficient $e_1 \big ( {\mathfrak q}_0, H^i_{R_+}(M)_n \big )$ of the $n$-th graded component of $H^i_{R_+}(M)$ with respect to ${\mathfrak q}_0$ is antipolynomial of degree $< i$ in $n$. In addition, we prove that the postulation numbers of the components $H^i_{R_+} (M)_n$ with respect to ${\mathfrak q}_0$ have a common upper bound.## References

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## Bibliographic Information

**M. Brodmann**- Affiliation: Institute of Mathematics, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
- MR Author ID: 41830
- 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
- Received by editor(s): December 1, 2003
- Published electronically: November 19, 2004
- Communicated by: Bernd Ulrich
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2117198