摘要
Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the co ondition CI, and N be a finitely generated R-module with SuppRN= V（a） for some a ∈ W（I, J）. It is shown that if ExtR^J（N,HI^i,J（M）） ∈ S for all i 〈 t and all j 〈 t - i, then Ha^i（M） ∈ S for all i 〈 t. Let S be the class of all R-modules N with divmR N ≤ k, where k is an integer. It is proved that if Ha^i（M） ∈ S for all i 〈 t and all a ∈ W（I, J）, then HI^i,j（M） ∈ S for all i 〈 t. It follows that inf{i ： HI^i,j（M） S} = inf（inf{i ： Ha^i（M） S）： a ∈W（I,J）}.
Let R be a Noetherian ring, I and J two ideals of R, M an R-module and t an integer. Let S be a Serre subcategory of the category of R-modules satisfying the co ondition CI, and N be a finitely generated R-module with SuppRN= V（a） for some a ∈ W（I, J）. It is shown that if ExtR^J（N,HI^i,J（M）） ∈ S for all i 〈 t and all j 〈 t - i, then Ha^i（M） ∈ S for all i 〈 t. Let S be the class of all R-modules N with divmR N ≤ k, where k is an integer. It is proved that if Ha^i（M） ∈ S for all i 〈 t and all a ∈ W（I, J）, then HI^i,j（M） ∈ S for all i 〈 t. It follows that inf{i ： HI^i,j（M） S} = inf（inf{i ： Ha^i（M） S）： a ∈W（I,J）}.
