Finiteness of certain local cohomology modules

Document Type : Research Paper

Author

Department of mathematics, Islamic Azad University Yazd Branch, Yazd, Iran.

10.29252/as.2020.1683

Abstract

Cofiniteness of the generalized local cohomology modules
$H^{i}_{\mathfrak{a}}(M,N)$ of two $R$-modules $M$ and $N$ with
respect to an ideal $\mathfrak{a}$ is studied for some $i^{,}s$ with
a specified property. Furthermore, Artinianness of
$H^{j}_{\mathfrak{b}_{0}}(H_{\mathfrak{a}}^{i}(M,N))$ is
investigated by using the above result, in certain graded situations, where $\mathfrak{b}_{0}$ is an ideal of $R_{0}$ and
$\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ such that
$\mathfrak{b}_{0}+\mathfrak{a}_{0}$ is an  $\mathfrak{m}_{0}$-primary ideal.

Keywords

References

[1] M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J.21(1980), 173-181.
[2] M. P. Brodmann, R. Y. Sharp, Local cohomology: An Algebraic introduction with geometric applications, Cambridge Univ. Press 1998, zb1 0903.13006.
[3] W. Bruns and J. Herzog, Cohen Macaulay rings, Cambridge Univ. Press 1993.
[4] F. Dehghani-Zadeh, On the niteness properties of generalized local cohomology modules, Int. Electronic. J. Algebra.
[5] J. Herzog, Komplexe, Auosungen und Dualitat in der Lokalen Algebra, Habilitationsschrift, Universitat Regensburg, 1974, htt://dx.doi.org/.
[6] A. Ma , Some results on local cohomology modules, Arch. Math. 87(2006), 211-216.
[7] H. Matsumura, Commutative ring theory. Cambridge Univ.Press 1986.
[8] T. Marley and J. Vassilev, Co niteness and associated primes of local cohomology modules, J. Alegebra. 256(2002) 180-193.
[9] L. Melkersson, On asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Cambridge Philos. Soc. 107(1990), 267-271.
[10] L. Melkersson, Properties of co nite modules and applications to local cohomology, Math. Proc. CambridgePhil. Soc. 125(1999), 417-423.
[11] L. T. Nhan. On generalized regular sequence and the niteness for associated primes of local cohomologymodules, Comm. Algebra. 33(2005), 793-806.
[12] J. Rotman, An introduction to homological algebra, 2nd edn. (Springer, 2009).
[13] N. Zamani, Finiteness Results on Generalized Local Cohomology Modules, Colloq. Math. 16(2009), no. 1, 65-70.
Algebraic Structures and Their Applications
Volume 7, Issue 1
Winter and Spring 2020
Pages 29-40

Files

Share

How to cite

Statistics