On the annihilators of formal local cohomology modules

Let a denote an ideal in a commutative Noetherian local ring (R, m) and M a non-zero finitely generated R-module of dimension d. Let d := dim(M / aM). In this paper we calculate the annihilator of the top formal local cohomology module F-a(d) (m). In fact, we prove that Ann(R) (F-a(d) (M)) = Ann(R) (M / U (R)(a, M)), where U-R(a, M) := boolean OR{N : N <= M and dim(N / aN) < dim(M / aM)}. We give a description of U-R (a, M) and we will show that Ann(R)(F-a(d) (M)) = Ann(R) (M / boolean AND p (j is an element of Assh RM boolean AND V(a)) N (j)), where 0 = boolean AND(n)(j=1) N-j denotes a reduced primary decomposition of the zero submodule 0 in M and N-j is a p(j)-primary submodule of M, for all j = 1, ..., n. Also, we determine the radical of the annihilator of F-a(d)(M). We will prove that root Ann(R) (F-a(d) (M)) Ann(R) (M / G(R) (a, M)), where G(R) (a, M) denotes the largest submodule of M such that Assh(R) (M) boolean AND V(a) subset of Ass(R) (M / G(R) (a, M)) and Assh(R) (M) denotes the set {p is an element of AssM : dim R/p = dim M}.