Annihilators and attached primes of local cohomology modules with respect to a system of ideals

Let \Phi be a system of ideals of a commutative Noetherian ring, we study the annihilators and attached primes of local cohomology modules with respect to a system of ideals.  We prove that if M is a non-zero finitely generated R-module of finite dimension d and \Phi is a system of ideals, thenAtt(H^d_\Phi(M))=\{p\in Ass M\mid cd(\Phi,R/p)=d\}. Moreover, if the cohomology dimension of M with respect to \Phi is dim M-۱, then Att(H^{dim M-۱}_\Phi(M))=\{p\in Supp M \mid cd(\Phi,R/p)=\dim M-۱\}.

Keywords:

Annihilators , Attached primes , cohomological dimension , Local cohomology