Top local cohomology modules with specified attached primes

Let (R,m) be a complete Noetherian local ring and let M be a finite R--module of positive Krull dimension n. It is shown that any subset T of Assh_R(M) can be expressed as the set of attached primes of the top local cohomology module H^n_a(M) for some ideal a of R. Moreover if a is an ideal of R such that the set of attached primes of H^n_a(M) is a non--empty proper subset of Assh_R(M), then H^n_a(M)=H^n_b(M) for some ideal b of R with dim_R (R/b)=1.