Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules

Document Type : Original Paper

Author

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

Abstract

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\fa$, is non-zero. $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\mathfrak a$, is zero for all $i$ with $i>p+c$. In this paper, we obtain the coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we obtain a necessary and sufficient condition for the vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules.

Highlights

1.M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21(2) (1980) 173–181.
2. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics 60 (Cambridge University Press, Cambridge, 1998).
3.W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39 (Cambridge University Press, Cambridge, 1993).
4.M. T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal, Arch. Math. (Basel) 84(4) (2005) 292–297.
5. K. Divaani-Aazar, Vanishing of the top local cohomology modules over Noetherian rings, Proc. Indian Acad. Sci. Math. Sci. 119(1) (2009) 23–35.
6.K. Divaani-Aazar and A. Hajikarimi, Generalized local cohomology modules and homological Gorenstein dimensions, Comm. Algebra 39(6) (2011) 2051–2067.
7.K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc. 130(12) (2002) 3537–3544.
8. K. Divaani-Aazar, R. Sazeedeh and M. Tousi, On vanishing of generalized local cohomology modules, Algebra Colloq. 12(2) (2005) 213–218.
9.E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Volume 1. Second revised and extended edition. De Gruyter Expositions in Mathematics 30 (Walter de Gruyter GmbH & Co. KG, Berlin, 2011).
10. A. Fathi, A. Tehranian and H. Zakeri, Filter regular sequences and generalized local cohomology modules, Bull. Malays. Math. Sci. Soc. 38(2) (2015) 467–482.
11.Y. Gu and L. Chu, Attached primes of the top generalized local cohomology modules, Bull. Aust. Math. Soc. 79(1) (2009) 59–67.
12.S. H. Hassanzadeh and A. Vahidi, On vanishing and cofiniteness of generalized local cohomology modules, Comm. Algebra 37(7) (2009) 2290–2299.
13. J. Herzog, Komplexe, Auflösungen und Dualität in der Localen Algebra (Habilitationsschrift, Universität Regensburg, 1970).
14. I. G. Macdonald, Secondary representation of modules over a commutative ring, Symp. Math. 11 (1973) 23–43.

15. A. Mafi, On the associated primes of generalized local cohomology modules, Comm. Algebra 34(7) (2006) 2489–2494.
16. H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8 (Cambridge University Press, Cambridge, 1986).
17. J. J. Rotman, An Introduction to Homological Algebra, Second edition, Universitext (Springer, New York, 2009).
18.S. Yassemi, Coassociated primes, Comm. Algebra 23(4) (1995) 1473–1498.

 

Keywords

Main Subjects

References

1.M. H. Bijan-Zadeh, A common generalization of local cohomology theories, Glasgow Math. J. 21(2) (1980) 173–181.
2. M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics 60 (Cambridge University Press, Cambridge, 1998).
3.W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39 (Cambridge University Press, Cambridge, 1993).
4.M. T. Dibaei and S. Yassemi, Attached primes of the top local cohomology modules with respect to an ideal, Arch. Math. (Basel) 84(4) (2005) 292–297.
5. K. Divaani-Aazar, Vanishing of the top local cohomology modules over Noetherian rings, Proc. Indian Acad. Sci. Math. Sci. 119(1) (2009) 23–35.
6.K. Divaani-Aazar and A. Hajikarimi, Generalized local cohomology modules and homological Gorenstein dimensions, Comm. Algebra 39(6) (2011) 2051–2067.
7.K. Divaani-Aazar, R. Naghipour and M. Tousi, Cohomological dimension of certain algebraic varieties, Proc. Amer. Math. Soc. 130(12) (2002) 3537–3544.
8. K. Divaani-Aazar, R. Sazeedeh and M. Tousi, On vanishing of generalized local cohomology modules, Algebra Colloq. 12(2) (2005) 213–218.
9.E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Volume 1. Second revised and extended edition. De Gruyter Expositions in Mathematics 30 (Walter de Gruyter GmbH & Co. KG, Berlin, 2011).
10. A. Fathi, A. Tehranian and H. Zakeri, Filter regular sequences and generalized local cohomology modules, Bull. Malays. Math. Sci. Soc. 38(2) (2015) 467–482.
11.Y. Gu and L. Chu, Attached primes of the top generalized local cohomology modules, Bull. Aust. Math. Soc. 79(1) (2009) 59–67.
12.S. H. Hassanzadeh and A. Vahidi, On vanishing and cofiniteness of generalized local cohomology modules, Comm. Algebra 37(7) (2009) 2290–2299.
13. J. Herzog, Komplexe, Auflösungen und Dualität in der Localen Algebra (Habilitationsschrift, Universität Regensburg, 1970).
14. I. G. Macdonald, Secondary representation of modules over a commutative ring, Symp. Math. 11 (1973) 23–43.
15. A. Mafi, On the associated primes of generalized local cohomology modules, Comm. Algebra 34(7) (2006) 2489–2494.
16. H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8 (Cambridge University Press, Cambridge, 1986).
17. J. J. Rotman, An Introduction to Homological Algebra, Second edition, Universitext (Springer, New York, 2009).
18.S. Yassemi, Coassociated primes, Comm. Algebra 23(4) (1995) 1473–1498.
Journal of Advanced Mathematical Modeling
Volume 13, Issue 2 - Serial Number 30
September 2023
Pages 250-258

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History

  • Receive Date: 20 May 2023
  • Revise Date: 30 July 2023
  • Accept Date: 02 August 2023

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