On the attached prime ideals of localcohomology modules defined by a pair of ideals

Abstract

Let I and J be two ideals of a commutative Noetherian ring R and M be an R -module of dimension d . For each i ∈ N0 let H i I,J (−) denote the i -th right derived functor of ΓI,J (−) , where ΓI,J (M ) := {x ∈ M : I n x ⊆ Jx for n ≫ 1} . If R is a complete local ring and M is finite, then attached prime ideals of H d−1 I,J (M ) are computed by means of the concept of co-localization. Moreover, we illustrate the attached prime ideals of H t I,J (M ) on a nonlocal ring R , for t = dim M and t = cd (I, J, M ) , where cd (I, J, M ) is the last nonvanishing level of H i I,J (M ) .