We study the finiteness of the sets $\operatorname{Ass}(H^d_{I, J}(M))$ and $\operatorname{Ass}(\operatorname{Hom}_R(R/I, H^d_{I, J}(M)))$ concerning Grothendieck's conjecture. We also show some properties of local cohomology modules $H^i_{I, J}(M)$ from the point of view of Serre subcategories.
References
M. Asgharzadeh and M. Tousi, A unified approach to local cohomology modules using Serre classes, Canad. Math. Bull. 53 (2010), no. 4, 577–586. 10.4153/cmb-2010-064-0 MR2761361 1205.13023 M. Asgharzadeh and M. Tousi, A unified approach to local cohomology modules using Serre classes, Canad. Math. Bull. 53 (2010), no. 4, 577–586. 10.4153/cmb-2010-064-0 MR2761361 1205.13023
K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (2005), no. 3, 655–660. MR2113911 1103.13010 10.1090/S0002-9939-04-07728-7 K. Divaani-Aazar and A. Mafi, Associated primes of local cohomology modules, Proc. Amer. Math. Soc. 133 (2005), no. 3, 655–660. MR2113911 1103.13010 10.1090/S0002-9939-04-07728-7
––––, Associated primes of local cohomology modules of weakly Laskerian modules, Comm. Algebra 34 (2006), no. 2, 681–690. 10.1080/00927870500387945 MR2211948 1097.13021 ––––, Associated primes of local cohomology modules of weakly Laskerian modules, Comm. Algebra 34 (2006), no. 2, 681–690. 10.1080/00927870500387945 MR2211948 1097.13021
A. Grothendieck, Cohomologie local des faisceaux coherents et theorèmes de Lefschetz locaux et globaux (SGA2), North-Holland, Amsterdam, 1968. 0159.50402 A. Grothendieck, Cohomologie local des faisceaux coherents et theorèmes de Lefschetz locaux et globaux (SGA2), North-Holland, Amsterdam, 1968. 0159.50402
J. J. Rotman, An Introduction to Homological Algebra, Second edition, Universitext, Springer, New York, 2009. 10.1007/978-0-387-68324-9 1157.18001 J. J. Rotman, An Introduction to Homological Algebra, Second edition, Universitext, Springer, New York, 2009. 10.1007/978-0-387-68324-9 1157.18001
R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra 213 (2009), no. 4, 582–600. 10.1016/j.jpaa.2008.09.008 MR2483839 1160.13013 R. Takahashi, Y. Yoshino and T. Yoshizawa, Local cohomology based on a nonclosed support defined by a pair of ideals, J. Pure Appl. Algebra 213 (2009), no. 4, 582–600. 10.1016/j.jpaa.2008.09.008 MR2483839 1160.13013
A. Tehranian and A. Pour Eshmanan Talemi, Cofiniteness of local cohomology based on a non-closed support defined by a pair of ideals, Bull. Iranian Math. Soc. 36 (2010), no. 2, 145–155. 1223.13008 A. Tehranian and A. Pour Eshmanan Talemi, Cofiniteness of local cohomology based on a non-closed support defined by a pair of ideals, Bull. Iranian Math. Soc. 36 (2010), no. 2, 145–155. 1223.13008
N. Tu Cuong and N. Van Hoang, On the vanishing and the finiteness of supports of generalized local cohomology modules, Manuscripta Math. 126 (2008), no. 1, 59–72. 10.1007/s00229-007-0162-7 MR2395249 1140.13015 N. Tu Cuong and N. Van Hoang, On the vanishing and the finiteness of supports of generalized local cohomology modules, Manuscripta Math. 126 (2008), no. 1, 59–72. 10.1007/s00229-007-0162-7 MR2395249 1140.13015
N. Zamani, Results on local cohomology of weakly Laskerian modules, J. Algebra Appl. 10 (2011), no. 2, 303–308. 10.1142/s0219498811004586 MR2795739 1222.13016 N. Zamani, Results on local cohomology of weakly Laskerian modules, J. Algebra Appl. 10 (2011), no. 2, 303–308. 10.1142/s0219498811004586 MR2795739 1222.13016
