Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-06T22:03:37.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Artinian and Non-Artinian Local Cohomology Modules

Published online by Cambridge University Press:  20 November 2018

Mohammad T. Dibaei  and
Alireza Vahidi
Mohammad T. Dibaei
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran, andSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Irane-mail: dibaeimt@ipm.ir
Alireza Vahidi
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran, Iran, andPayame Noor University (PNU), Irane-mail: vahidi.ar@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\mathfrak{a},\,\mathfrak{b},\,\mathfrak{a}\,\cap \,\mathfrak{b}$ and $\mathfrak{a}\,+\,\mathfrak{b}$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen–Macaulay if there exists an ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer such that $0\,\le \,r\,<\,{{\dim}_{R}}(M)$, any maximal element q of the non-empty set of ideals {$\mathfrak{a}\,:\,\text{H}_{\mathfrak{a}}^{i}(M)$ is not artinian for some $i$, $i\,\ge \,r$} is a prime ideal, and all Bass numbers of $\text{H}_{\mathfrak{q}}^{i}(M)$ are finite for all $i\,\ge \,r$.

Keywords

13D4513E10local cohomology modulescohomological dimensionsBass numbers
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 4 , 01 December 2011 , pp. 619 - 629
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Aghapournahr, M. and Melkersson, L., Local cohomology and Serre subcategories. J. Algebra 320(2008), no. 3, 1275–1287. doi:10.1016/j.jalgebra.2008.04.002Google Scholar
[2] Bahmanpour, K. and Naghipour, R., Associated primes of local cohomology modules and Matlis duality. J. Algebra 320(2008), no. 6, 2632–2641. doi:10.1016/j.jalgebra.2008.05.014Google Scholar
[3] Brodmann, M. P., Fumasoli, S., and Tajarod, R., Local cohomology over homogeneous rings with one-dimensional local base ring. Proc. Amer. Math. Soc. 131(2003), no. 10, 2977–2985. doi:10.1090/S0002-9939-03-07009-6Google Scholar
[4] Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics, 60, Cambridge University Press, Cambridge, 1998.Google Scholar
[5] Delfino, D. and Marley, T., Cofinite modules and local cohomology. J. Pure Appl. Algebra 121(1997), no. 1, 45–52. doi:10.1016/S0022-4049(96)00044-8Google Scholar
[6] Dibaei, M. T. and Yassemi, S., Cohomological dimension of complexes. Comm. Algebra 32(2004), no. 11, 4375–4386. doi:10.1081/AGB-200034165Google Scholar
[7] Dibaei, M. T. and Yassemi, S., Attached primes of the top local cohomology modules with respect to an ideal. Arch. Math. 84(2005), no. 4, 292–297.Google Scholar
[8] Dibaei, M. T. and Yassemi, S., Bass numbers of local cohomology modules with respect to an ideal. Algebr Represent. Theory 11(2008), no. 3, 299–306. doi:10.1007/s10468-007-9072-3Google Scholar
[9] Divaani-Aazar, K., Naghipour, R., and Tousi, M., Cohomological dimension of certain algebraic varieties. Proc. Amer. Math. Soc. 130(2002), no. 12, 3537–3544. doi:10.1090/S0002-9939-02-06500-0Google Scholar
[10] Hellus, M., Local cohomology and Matlis duality.. Univ. Iagel. Acta Math. 45(2007), 63–70.Google Scholar
[11] Huneke, C., Problems on local cohomology. In: Free resolutions in commutative algebra and algebraic geometry (Sundance, UT, 1990), Res. Notes Math., 2, Jones and Bartlett, Boston, MA, 1992, pp. 93–108.Google Scholar
[12] Macdonald, I. G., Secondary representation of modules over a commutative ring. In: Symposia Mathematica, XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), Academic Press, London, 1973, pp. 23–43.Google Scholar
[13] Macdonald, I. G. and Sharp, R. Y., An elementary proof of the non-vanishing of certain local cohomology modules.. Quart. J. Math. Oxford Ser. (2) 23(1972), 197–204. doi:10.1093/qmath/23.2.197Google Scholar
[14] Rotman, J. J., An introduction to homological algebra. Pure and Applied Mathematics, 85, Academic Press, New York-London, 1979.Google Scholar
[15] Zöschinger, H., Minimax-moduln. (German) J. Algebra 102(1986), no. 1, 1–32. doi:10.1016/0021-8693(86)90125-0Google Scholar