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 is finite whenever R is regular. Also, it is shown that if x1,…,xd is a system of parameters for R, then has infinitely many associated prime ideals for all i⩽d−1, where D(−):=HomR(−,E) denotes the Matlis dual functor and E:=ER(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 is not finitely generated, where I=(x1)∩(x2,…,xd).