An Example of an Infinite Set of Associated Primes of a Local Cohomology Module

Let (R,m) be a local Noetherian ring, let I ⊂ R be any ideal and let M be a finitely generated R-module. It has been long conjectured that the local cohomology modules H I(M) have finitely many associated primes for all i (see Conjecture 5.1 in [H] and [L].) If R is not required to be local these sets of associated primes may be infinite, as shown by Anurag Singh in [S], where he constructed an example of a local cohomology module of a finitely generated module over a finitely generated Z-algebra with infinitely many associated primes. This local cohomology module has p-torsion for all primes p ∈ Z. However, the question of the finiteness of the set of associated primes of local cohomology modules defined over local rings and over k-algebras (where k is a field) has remained open until now. In this paper I settle this question by constructing a local cohomology module of a local finitely generated k-algebra with an infinite set of associated primes, and I do this for any field k.