On annihilators and associated primes of local cohomology modules

Abstract

We establish the Local-global Principle for the annihilation of local cohomology modules over an arbitrary commutative Noetherian ring R at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 4. We explore interrelations between the principle and the Annihilator Theorem for local cohomology, and show that, if R is universally catenary and all formal fibres of all localizations of R satisfy Serre's condition ( S r ), then the Annihilator Theorem for local cohomology holds at level r over R if and only if the Local-global Principle for the annihilation of local cohomology modules holds at level r over R. Moreover, we show that certain local cohomology modules have only finitely many associated primes. This provides motivation for the study of conditions under which the set ⋃ m,n∈ N Ass(M/(x m,y n)M) (where M is a finitely generated R-module and x, y∈ R) is finite: an example due to M. Katzman shows that this set is not always finite; we provide some sufficient conditions for its finiteness.