Geodesic
Cofiniteness and finiteness of local cohomology modules over regular local rings
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 733-740
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/CMJ.2017.0116-16
Classification : 13D45, 13E05, 14B15
Keywords: cofinite module; Cohen-Macaulay ring; Krull dimension; local cohomology; regular ring 
@article{10_21136_CMJ_2017_0116_16,
     author = {A'zami, Jafar and Pourreza, Naser},
     title = {Cofiniteness and finiteness of local cohomology modules over regular local rings},
     journal = {Czechoslovak Mathematical Journal},
     pages = {733--740},
     year = {2017},
     volume = {67},
     number = {3},
     doi = {10.21136/CMJ.2017.0116-16},
     mrnumber = {3697912},
     zbl = {06770126},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0116-16/}
}
TY  - JOUR
AU  - A'zami, Jafar
AU  - Pourreza, Naser
TI  - Cofiniteness and finiteness of local cohomology modules over regular local rings
JO  - Czechoslovak Mathematical Journal
PY  - 2017
SP  - 733
EP  - 740
VL  - 67
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0116-16/
DO  - 10.21136/CMJ.2017.0116-16
LA  - en
ID  - 10_21136_CMJ_2017_0116_16
ER  - 
%0 Journal Article
%A A'zami, Jafar
%A Pourreza, Naser
%T Cofiniteness and finiteness of local cohomology modules over regular local rings
%J Czechoslovak Mathematical Journal
%D 2017
%P 733-740
%V 67
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0116-16/
%R 10.21136/CMJ.2017.0116-16
%G en
%F 10_21136_CMJ_2017_0116_16
A'zami, Jafar; Pourreza, Naser. Cofiniteness and finiteness of local cohomology modules over regular local rings. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 733-740. doi: 10.21136/CMJ.2017.0116-16

[1] Bagheriyeh, I., Bahmanpour, K., A'zami, J.: Cofiniteness and non-vanishing of local cohomology modules. J. Commut. Algebra 6 (2014), 305-321. | DOI | MR | JFM

[2] Bahmanpour, K.: Annihilators of local cohomology modules. Commun. Algebra 43 (2015), 2509-2515. | DOI | MR | JFM

[3] Bahmanpour, K., A'zami, J., Ghasemi, G.: On the annihilators of local cohomology modules. J. Algebra 363 (2012), 8-13. | DOI | MR | JFM

[4] Bahmanpour, K., Naghipour, R.: Associated primes of local cohomology modules and Matlis duality. J. Algebra 320 (2008), 2632-2641. | DOI | MR | JFM

[5] Bahmanpour, K., Naghipour, R.: Cofiniteness of local cohomology modules for ideals of small dimension. J. Algebra 321 (2009), 1997-2011. | DOI | MR | JFM

[6] Brodmann, M. P., Sharp, R. Y.: Local Cohomology. An Algebraic Introduction with Geometric Applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge (1998). | DOI | MR | JFM

[7] Grothendieck, A.: 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), French. | MR | JFM

[8] Hartshorne, R.: Affine duality and cofiniteness. Invent. Math. 9 (1970), 145-164. | DOI | MR | JFM

[9] Hellus, M.: On the set of associated primes of a local cohomology module. J. Algebra 237 (2001), 406-419. | DOI | MR | JFM

[10] Huneke, C., Koh, J.: Cofiniteness and vanishing of local cohomology modules. Math. Proc. Camb. Philos. Soc. 110 (1991), 421-429. | DOI | MR | JFM

[11] Khashyarmanesh, K.: On the finiteness properties of extension and torsion functors of local cohomology modules. Proc. Am. Math. Soc. 135 (2007), 1319-1327. | DOI | MR | JFM

[12] Khashyarmanesh, K., Salarian, Sh.: Filter regular sequences and the finiteness of local cohomology modules. Commun. Algebra 26 (1998), 2483-2490. | DOI | MR | JFM

[13] Melkersson, L.: Modules cofinite with respect to an ideal. J. Algebra 285 (2005), 649-668. | DOI | MR | JFM

[14] Schenzel, P.: Proregular sequences, local cohomology, and completion. Math. Scand. 92 (2003), 161-180. | DOI | MR | JFM

Cité par Sources :