Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 733-740, 2017


Cofiniteness and finiteness of local cohomology modules over regular local rings

Jafar A'zami, Naser Pourreza

Received March 8, 2016.  First published March 27, 2017.

Abstract:  Let $(R,\mathfrak m)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\rm mAss}_R(R/I)={\rm Assh}_R(I)$. It is shown that the $R$-module $H^{{\rm ht}(I)}_I(R)$ is $I$-cofinite if and only if ${\rm cd}(I,R)={\rm ht}(I)$. Also we present a sufficient condition under which this condition the $R$-module $H^i_I(R)$ is finitely generated if and only if it vanishes.
Keywords:  cofinite module; Cohen-Macaulay ring; Krull dimension; local cohomology; regular ring
Classification MSC:  13D45, 14B15, 13E05


References:
[1] I. Bagheriyeh, K. Bahmanpour, J. A'zami: Cofiniteness and non-vanishing of local cohomology modules. J. Commut. Algebra 6 (2014), 305-321. DOI 10.1216/JCA-2014-6-3-305 | MR 3278806 | Zbl 1299.13019
[2] K. Bahmanpour: Annihilators of local cohomology modules. Commun. Algebra 43 (2015), 2509-2515. DOI 10.1080/00927872.2014.900687 | MR 3344203 | Zbl 1323.13003
[3] K. Bahmanpour, J. A'zami, G. Ghasemi: On the annihilators of local cohomology modules. J. Algebra 363 (2012), 8-13. DOI 10.1016/j.jalgebra.2012.03.026 | MR 2925842 | Zbl 1262.13027
[4] K. Bahmanpour, R. Naghipour: Associated primes of local cohomology modules and Matlis duality. J. Algebra 320 (2008), 2632-2641. DOI 10.1016/j.jalgebra.2008.05.014 | MR 2441778 | Zbl 1149.13008
[5] K. Bahmanpour, R. Naghipour: Cofiniteness of local cohomology modules for ideals of small dimension. J. Algebra 321 (2009), 1997-2011. DOI 10.1016/j.jalgebra.2008.12.020 | MR 2494753 | Zbl 1168.13016
[6] M. P. Brodmann, R. Y. Sharp: Local Cohomology. An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511629204 | MR 1613627 | Zbl 0903.13006
[7] A. Grothendieck: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz loceaux et globeaux (SGA 2). Séminaire de Géométrie Algébrique du Bois-Marie, 1962, Advanced Studies in Pure Mathematics 2, North-Holland Publishing Company, Amsterdam; Masson & Cie, Éditeur, Paris (1968). (In French.) MR 0476737 | Zbl 0197.47202
[8] R. Hartshorne: Affine duality and cofiniteness. Invent. Math. 9 (1970), 145-164. DOI 10.1007/BF01404554 | MR 0257096 | Zbl 0196.24301
[9] M. Hellus: On the set of associated primes of a local cohomology module. J. Algebra 237 (2001), 406-419. DOI 10.1006/jabr.2000.8580 | MR 1813886 | Zbl 1027.13009
[10] C. Huneke, J. Koh: Cofiniteness and vanishing of local cohomology modules. Math. Proc. Camb. Philos. Soc. 110 (1991), 421-429. DOI 10.1017/S0305004100070493 | MR 1120477 | Zbl 0749.13007
[11] K. Khashyarmanesh: On the finiteness properties of extension and torsion functors of local cohomology modules. Proc. Am. Math. Soc. 135 (2007), 1319-1327. DOI 10.1090/S0002-9939-06-08664-3 | MR 2276640 | Zbl 1111.13016
[12] K. Khashyarmanesh, Sh. Salarian: Filter regular sequences and the finiteness of local cohomology modules. Commun. Algebra 26 (1998), 2483-2490. DOI 10.1080/00927879808826293 | MR 1627876 | Zbl 0909.13007
[13] L. Melkersson: Modules cofinite with respect to an ideal. J. Algebra 285 (2005), 649-668. DOI 10.1016/j.jalgebra.2004.08.037 | MR 2125457 | Zbl 1093.13012
[14] P. Schenzel: Proregular sequences, local cohomology, and completion. Math. Scand. 92 (2003), 161-180. DOI 10.7146/math.scand.a-14399 | MR 1973941 | Zbl 1023.13011

Affiliations:   Jafar A'zami (corresponding author), Naser Pourreza, Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Daneshgah Street, Ardabil, 56199-11367, Iran, e-mail: jafar.azami@gmail.com, azami@uma.ac.ir, pourreza1974@gmail.com


 
PDF available at: