Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show th...Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show that inf{f-depth（a, M） 丨a ∈ W（I, J）} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H_I,J（M） is （I, J）-cofinite for all i 〈 inf{f-depth（a, M） 丨a ∈ W（I, J）}. In addition, we show that for a Serre subcategory S, if H_I,J（M） belongs to S for all i 〉 n and if b is an ideal of R such that H^n_I,J（M/bM） belongs to S, then the module H^n_I,J（M）/bH^n_I,J（M） belongs to S.展开更多
文摘Let I, J be ideals of a commutative Noetherian local ring （R, m） and let M be a finite R-module. The f-depth of M with respect to I is the least integer r such that H_I（M） is not Artinian. In this paper we show that inf{f-depth（a, M） 丨a ∈ W（I, J）} is the least integer such that the local cohomology module with respect to a pair of ideals I, J is not Artinian. As a consequence, it follows that H_I,J（M） is （I, J）-cofinite for all i 〈 inf{f-depth（a, M） 丨a ∈ W（I, J）}. In addition, we show that for a Serre subcategory S, if H_I,J（M） belongs to S for all i 〉 n and if b is an ideal of R such that H^n_I,J（M/bM） belongs to S, then the module H^n_I,J（M）/bH^n_I,J（M） belongs to S.
