Associated primes of local cohomology modules and Matlis duality

Abstract

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 I 2 ( R ) is finite whenever R is regular. Also, it is shown that if x 1 , … , x d is a system of parameters for R, then D ( H ( x 1 , … , x i ) 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 I d − 1 ( R ) ) is not finitely generated, where I = ( x 1 ) ∩ ( x 2 , … , x d ) .