## Artinianess of graded local cohomology modules

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- by Reza Sazeedeh PDF
- Proc. Amer. Math. Soc.
**135**(2007), 2339-2345 Request permission

## Abstract:

Let $R=\bigoplus _{n \in \mathbb {N}} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak {m}_0)$ and let $M$ be a finitely generated graded $R$-module. Let $a$ be the largest integer such that $H_{R_+}^a(M)$ is not Artinian. We will prove that $H_{R_+}^i(M)/\mathfrak {m}_0H_{R_+}^i(M)$ are Artinian for all $i\geq a$ and there exists a polynomial $\widetilde {P}\in \mathbb {Q}[\mathbf {x}]$ of degree less than $a$ such that $\textrm {length}_{R_0}(H_{R_+}^a(M)_n /\mathfrak {m}_0H_{R_+}^a(M)_n) =\widetilde {P}(n)$ for all $n\ll 0$. Let $s$ be the first integer such that the local cohomology module $H_{R_+}^s(M)$ is not ${R_+}-$cofinite. We will show that for all $i\leq s$ the graded module $\Gamma _{\mathfrak {m}_0}(H_{R_+}^i(M))$ is Artinian.## References

- 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**, DOI 10.1017/CBO9780511629204 - 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**, DOI 10.1090/S0002-9939-03-07009-6 - M. Brodmann, F. Rohrer, and R. Sazeedeh,
*Multiplicities of graded components of local cohomology modules*, J. Pure Appl. Algebra**197**(2005), no. 1-3, 249–278. MR**2123988**, DOI 10.1016/j.jpaa.2004.08.034 - W. Bruns and J. Herzog,
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics 39, Revised edition, Cambridge University Press (1998). - Mohammad T. Dibaei and Siamak Yassemi,
*Associated primes and cofiniteness of local cohomology modules*, Manuscripta Math.**117**(2005), no. 2, 199–205. MR**2150481**, DOI 10.1007/s00229-005-0538-5 - D. Kirby,
*Artinian modules and Hilbert polynomials*, Quart. J. Math. Oxford Ser. (2)**24**(1973), 47–57. MR**316446**, DOI 10.1093/qmath/24.1.47 - Mordechai Katzman and Rodney Y. Sharp,
*Some properties of top graded local cohomology modules*, J. Algebra**259**(2003), no. 2, 599–612. MR**1955534**, DOI 10.1016/S0021-8693(02)00567-7 - Christel Rotthaus and Liana M. Şega,
*Some properties of graded local cohomology modules*, J. Algebra**283**(2005), no. 1, 232–247. MR**2102081**, DOI 10.1016/j.jalgebra.2004.07.034

## Additional Information

**Reza Sazeedeh**- Affiliation: Department of Mathematics, Urmia University, Urmia, Iran –and– Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran
- Email: rsazeedeh@ipm.ir
- Received by editor(s): January 9, 2006
- Received by editor(s) in revised form: April 6, 2006
- Published electronically: March 21, 2007
- Additional Notes: This research was in part supported by a grant from IPM (No. 84130033)
- Communicated by: Bernd Ulrich
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**135**(2007), 2339-2345 - MSC (2000): Primary 13D45, 13E10
- DOI: https://doi.org/10.1090/S0002-9939-07-08794-1
- MathSciNet review: 2302554