Hostname: page-component-cb9f654ff-mx8w7 Total loading time: 0 Render date: 2025-09-01T23:07:53.965Z Has data issue: false hasContentIssue false
  • English
  • Français

Artinianness of Composed Graded Local Cohomology Modules

Published online by Cambridge University Press:  20 November 2018

Fatemeh Dehghani-Zadeh
Fatemeh Dehghani-Zadeh*
Affiliation:
Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd, Iran e-mail: fdzadeh@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 $R\,=\,{{\oplus }_{n\ge 0}}{{R}_{n}}$ be a graded Noetherian ring with local base ring $\left( {{R}_{0}},{{\text{m}}_{0}} \right)$ and let ${{R}_{+}}\,=\,{{\oplus }_{n>0}}{{R}_{n}}$ . Let $M$ and $N$ be finitely generated graded $R$ -modules and let $\mathfrak{a}\,=\,{{\mathfrak{a}}_{0}}\,+\,{{R}_{+}}$ an ideal of $R$ . We show that $H_{\mathfrak{b}0}^{j}\,\left( H_{\mathfrak{a}}^{i}\left( M,\,N \right) \right)$ and ${H_{\mathfrak{a}}^{i}\left( M,\,N \right)}/{{{\mathfrak{b}}_{0}}H_{\mathfrak{a}}^{i}\left( M,\,N \right)}\;$ are Artinian for some $i\text{ s}$ and $j\,\text{s}$ with a specified property, where ${{\mathfrak{b}}_{o}}$ is an ideal of ${{R}_{0}}$ such that ${{\mathfrak{a}}_{0}}\,+\,{{\mathfrak{b}}_{0}}$ is an ${{\mathfrak{m}}_{0}}$ -primary ideal.

Keywords

13D4513E1016W50generalized local cohomologyArtinian, graded module

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 271 - 278
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Brodmann, M. P., Asymptotic behaviour of cohomology: tameness,supports and associated primes. In: Commutative algebra and algebraic geometry, Contemp. Math., 390, American Mathematical Society, Providence, RI, 2005, pp. 31–61. http://dx.doi.org/10.1090/conm/390/07292 Google Scholar
[2] Brodmann, M. P. and Hellus, M., Cohomological pattern of coherent sheaves over protective schemes. J. Pure Appl. Algebra 172(2002), no. 2-3, 165–182. http://dx.doi.org/10.1016/S0022-4049(01)00144-X Google Scholar
[3] Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Mathematics, 60, Cambridge University Press, Cambridge, 1998. http://dx.doi.org/10.1017/CBO9780511629204 Google Scholar
[4] Bruns, W. and Herzog, J., Cohen-Macaulay rings. Cambridge Studies in Mathematics, 39, Cambridge University Press, Cambridge, 1993.Google Scholar
[5] Chu, L. and Tang, Z., On the Artinianness of generalized local cohomology. Comm. Algebra 35(2007), no 12, 3821–3827. http://dx.doi.org/10.1080/00927870701511517 Google Scholar
[6] Dehghani-Zadeh, F., Finiteness properties generalized local cohomology with respect to an ideal containing the irrelevant ideal. J. Korean Math. Soc. 49(2012), no. 6,1215-1227. http://dx.doi.org/10.4134/JKMS.2012.49.6.1215 Google Scholar
[7] Jahangiri, M., Shirmohammadi, N., and Tahamtan, Sh., Tameness and Artinianness of graded generalized local cohomology modules. Algebra Colloq. 22(2015), no. 1,131-146. http://dx.doi.org/10.1142/S100538671 5000127 Google Scholar
[8] Kirby, D., Artinian modules and Hilbertpolynomials. Quart. J. Math. Oxford Ser. (2) 24(1973), 47–57.Google Scholar
[9] Melkerson, L., Modules cofinite with respect to an ideal. J. Algebra, 285(2005), no. 2, 649–668. http://dx.doi.org/10.1016/j.jalgebra.2004.08.037 Google Scholar
[10] Melkerson, L., Properties of cofinite modules and applications to local cohomology. Math.Proc. Cambridge Philos. Soc. 125(1999), no. 3, 417–423. http://dx.doi.org/10.1017/S0305004198003041 Google Scholar
[11] Rotman, J. J., An introduction to homological algebra. Pure and Applied Mathematics, 85, Academic Press, New York-London, 1979.Google Scholar