Constructing arbitrary torsion elements for a local cohomology module

Abstract

Local cohomology modules are, in general, not finitely generated over the ambient ring. Towards a better understanding of the structure of these modules, Huneke (1992) [Hu] asked if they have finitely many associated prime ideals. The answer is negative. Several authors have constructed a ring whose local cohomology module has infinitely many associated primes. In certain situations, one can prove that a local cohomology module has infinitely many associated primes by showing that Z/pZ embeds into the local cohomology module for each prime integer p. We show that in fact, each finitely generated abelian group embeds into a local cohomology module.