Local cohomology modules and Cousin complexes

Document Type : Research Paper


Department of Mathematics, Payame Noor University (PNU), P.O.BOX, 19395-4697, Tehran, Iran



Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $X$ an arbitrary $R$--module, $\mathcal{F}$ a filtration of $\operatorname{Spec}(R)$ which admits $X$, and $s, s', t, t'$ non-negative integers such that $s+ t= s'+ t'$. In this paper, we study the membership of $R$--modules $\operatorname{H}^{s}_\mathfrak{a}(\operatorname{H}^{t- 1}(\operatorname{C}_R(\mathcal{F}, X)))$ and $\operatorname{H}^{s'- 1}(\operatorname{H}^{t'}_\mathfrak{a}(\operatorname{C}_R(\mathcal{F}, X)))$ in Serre subcategories of the category of $R$--modules and find some sufficient conditions which ensure the existence of an isomorphism between them, where $\operatorname{C}_R(\mathcal{F},X)$ is the Cousin complex for $X$ with respect to $\mathcal{F}$. As applications, we give some new facts and represent some older facts about the local cohomology modules and the Cousin complexes.


[1] M. Aghapournahr, L. Melkersson, Local cohomology and Serre subcategories, J. Algebra 320 (2008) 1275-1287.
[2] M. Asgharzadeh, M. Tousi, A uni ed approach to local cohomology modules using Serre classes, Canad. Math. Bull. 53 (2010) 577-586.
[3] H. Bamdad, A. Vahidi, Extension functors of Cousin cohomology modules, Bull. Iranian Math. Soc. 44 (2018) 253-267.
[4] M. P. Brodmann, R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.
[5] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1998.
[6] R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics 20, Springer, Berlin, 1966.
[7] R. Hartshorne, Cohomological dimension of algeraic varieties, Ann. of Math. 88 (1968) 403-450.
[8] C. Huneke, Problems on Local Cohomology: Free resolutions in Commutative Algebra and Algebraic Geometry, Jones and Bartlett, Boston, 1992.
[9] H. Petzl, Cousin complexes and at ring extensions, Comm. Algebra 25 (1997) 311-339.
[10] J. J. Rotman, An Introduction to Homological Algebra, Springer Science and Business Media, New York, 2008.
[11] R. Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969) 340-356.
[12] R. Y. Sharp, Gorenstein modules, Math. Z. 115 (1970) 117-139.
[13] R. Y. Sharp, Local cohomology and the Cousin complex for a commutative Noetherian ring, Math. Z. 153 (1977) 19-22.
[14] R. Y. Sharp, A Cousin complex characterization of balanced big Cohen-Macaulay modules, Q. J. Math. 33 (1982) 471-485.
[15] A. Vahidi, M. Aghapournahr, Some results on generalized local cohomology modules, Comm. Algebra 43 (2015) 2214-2230.
[16] H. Zoschinger, Minimax moduln, J. Algebra 102 (1986) 1-32.