On annihilators and associated primes of local cohomology modules

We establish the Local-global Principle for the annihilation of local cohom ology modules over an arbitrary commutative Noetherian ring R at level 2. W e also establish the same principle at all levels over an arbitrary commuta tive Noetherian ring of dimension not exceeding 4. We explore interrelation s between the principle and the Annihilator Theorem for local cohomology, a nd show that, if R is universally catenary and all formal fibres of all loc alizations of IZ satisfy Serre's condition (S-r), then the Annihilator Theo rem for local cohomology holds at level r over R if and only if the Local-g lobal Principle for the annihilation of local cohomology modules holds at l evel r over R. Moreover, we show that certain local cohomology modules have only finitely many associated primes. This provides motivation for the stu dy of conditions under which the set U(m,n)epsilon N ASS(M/(x(m), y(n))M) ( where M is a finitely generated R-module and x, y epsilon R) is finite: an example due to M. Katzman shows that this set is not always finite; we prov ide some sufficient conditions for its finiteness. (C) 2000 Elsevier Scienc e B.V. All rights reserved.