Finiteness of Graded Generalized Local Cohomology Modules

A. Mafi ; H. Saremi

Matematičeskie zametki, Tome 94 (2013) no. 5, pp. 689-694.

Résumé

We consider two finitely generated graded modules over a homogeneous Noetherian ring $R=\bigoplus_{n\in\mathbb{N}_0}R_n$ with a local base ring $(R_0,\mathfrak{m}_0)$ and irrelevant ideal $R_{+}$ of $R$. We study the generalized local cohomology modules $H_\mathfrak{b}^i(M,N)$ with respect to the ideal $\mathfrak{b}=\mathfrak{b}_0+{R}_+$, where $\mathfrak{b}_0$ is an ideal of $R_0$. We prove that if $\operatorname{dim} R_0/\mathfrak{b}_0\le 1$, then the following cases hold: for all $i\ge 0$, the $R$-module $H_\mathfrak{b}^i(M,N)/{\mathfrak{a}_0H_\mathfrak{b}^i(M,N)}$ is Artinian, where $\sqrt{\mathfrak{a}_0+\mathfrak{b}_0}=\mathfrak{m}_0$; for all $i\ge 0$, the set $\operatorname{Ass}_{R_0}(H_\mathfrak{b}^i(M,N)_n)$ is asymptotically stable as $n\to{-\infty}$. Moreover, if $H_{\mathfrak{b}}^j(M,N)_n$ is a finitely generated $R_0$-module for all $n\le n_0$ and all $j$, where $n_0\in\mathbb{Z}$ and $i\in\mathbb{N}_0$, then for all $n\le n_0$, the set $\operatorname{Ass}_{R_0}(H_{\mathfrak{b}}^i(M,N)_n)$ is finite.

Détail

Keywords: local cohomology modules, generalized local cohomology modules, graded modules, Noetherian ring.

Bibliographie

[1] M. P. Brodmann, R. Y. Sharp, Local Cohomology. An Algebraic Introduction with Geometric Applications, Cambridge Stud. Adv. Math., 60, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[2] M. Brodmann, M. Hellus, “Cohomological patterns of coherent sheaves over projective schemes”, J. Pure Appl. Algebra, 172:2-3 (2002), 165–182 | DOI | MR | Zbl

[3] M. Brodmann, “Asymptotic behaviour of cohomology: tameness, supports and associated primes”, Commutative Algebra and Algebraic Geometry, Contemp. Math., 390, Amer. Math. Soc., Providence, RI, 2005, 31–61 | MR | Zbl

[4] M. Brodmann, S. Fumasoli, R. Tajarod, “Local cohomology over homogeneous rings with one-dimensional local base ring”, Proc. Amer. Math. Soc., 131:10 (2003), 2977–2985 | DOI | MR | Zbl

[5] K. Khashyarmanesh, A. Abbasi, “On the asymptotic behaviour of associated primes of generalized local cohomology modules”, J. Aust. Math. Soc., 83:2 (2007), 217–225 | DOI | MR | Zbl

[6] C. Rotthaus, L. M. Şega, “Some properties of graded local cohomology modules”, J. Algebra, 283:1 (2005), 232–247 | DOI | MR | Zbl

[7] K. Khashyarmanesh, “Associated primes of graded components of generalized local cohomology modules”, Comm. Algebra, 33:9 (2005), 3081–3090 | MR | Zbl

[8] A. Mafi, “On minimax and graded local cohomology modules”, Southeast Asian Bull. Math., 35:5 (2011), 805–812 | MR | Zbl

[9] A. Mafi, H. Saremi, “Artinianness of certain graded local cohomology modules”, Canad. Math. Bull., 55:1 (2012), 153–156 | DOI | MR | Zbl

[10] M. Jahangiri, H. Zakeri, “Local cohomology modules with respect to an ideal containing the irrelevant ideal”, J. Pure Appl. Algebra, 213:4 (2009), 573–581 | DOI | MR | Zbl

[11] J. Herzog, Komplexe, auflösungen und dualitat in der localen algebra, Habilitationss chrift, Universität Regensburg, 1970

[12] N. Suzuki, “On the generalized local cohomology and its duality”, J. Math. Kyoto Univ., 18:1 (1978), 71–85 | MR | Zbl

[13] W. Bruns, J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math., 39, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[14] K. Divaani-Aazar, A. Hajikarimi, “Generalized local cohomology modules and homological Gorenstein dimensions”, Comm. Algebra, 39:6 (2011), 2051–2067 | DOI | MR | Zbl

[15] J. Amjadi, R. Naghipour, “Cohomological dimension of generalized local cohomology modules”, Algebra Colloq., 15:2 (2008), 303–308 | MR | Zbl

[16] A. Mafi, “Matlis reflexive and generalized local cohomology modules”, Czechoslovak Math. J., 59:4 (2009), 1095–1102 | DOI | MR | Zbl