Let R be a commutative Noetherian ring, M be a finitely generated R-module and a be an ideal of R such that aM , M. We show among the other things that, if c is a nonnegative integer such that Hia(M) = 0 for all i < c, then there is an isomorphism End(Hc a (M)) Extc R(Hc a (M); M); and if c is a nonnegative integer such that Hia (M) = 0 for all i , c, there are the following isomorphisms: (i) Hi b(Hc a (M)) Hi+c b (M) and (ii) Exti R(R=b; Hc a (M)) Exti+c R (R=b; M) for all i 2 N0 and all ideals b of R with b a. We also prove that if a and b are ideals of R with b a and c := grade(a; M), then there exists a natural homomorphism from End(Hc a (M)) to End(Hc b (M)), where grade(a; M) is the maximum length of M-sequences in a.
