Graded Local Cohomology Modules and Their Associated Primes#

Abstract

Let R = R 0[x 1, …, x t ] be a polynomial ring over R 0 and I be an ideal of R generated by a R-regular sequence of homogeneous elements. We wish to investigate the behaviour of the nth graded component of the ith local cohomology module of with respect to the irrelevant ideal R + = (x 1, …, x t ). Our two main results are: If dim R 0 = 1, then becomes constant when n becomes negatively large; if dim R 0 = 2, then there exists an integer N such that either for all n < N or, for all n < N. It should be noted, that does not become constant in general, even in the special case, where R 0 is regular local of dimension 4, t = 2 and I is principal [cf. Katzman, M. (2002). An example of an infinite set of associated primes of a local cohomology module. J. Algebra 252:161–166].