Associated primes of local cohomology modules and Matlis duality

Let (R, m) be a commutative Noetherian local ring of dimension d and I an ideal of R. We show that the set of associated primes of the local cohomology module H(l)(2)(R) is finite whenever R is regular. Also, it is shown that if X(l),...,x(d) is a system of parameters for R, then D(H((x,...,xi))(i)(R)) has infinitely many associated prime ideals for all i <= d-1, where D(-) := HOM(R)(-, E) denotes the Matlis dual functor and E := E(R)(R/m) is the injective hull of the residue field R/m. Finally, we explore a counterexample of Grothendieck's conjecture by showing that, if d >= 3, then the R-module HOM(R) (R/I, H(l)(d-1)(R)) is not finitely generated, where I = (x(l)) boolean AND (x(2),...,x(d)). (C) 2008 Elsevier Inc. All rights reserved.