Graded local cohomology modules and their associated primes

摘要

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 H-R+(i) (R/I)(n) of the ith local cohomology module of R/I with respect to the irrelevant ideal R+ = (x(1),. x(t)). Our two I main results are: If dim R-0 = 1, then Ass(R0) (H-R+(i) (R/I)(n)) becomes constant when n becomes negatively large; if dim R-0 = 2, then there exists an integer N such that either H-R+(i) (R/I)(n) = (0) for all n < N or, H-R+(i) (R/I) not equal (0) for all n < N. It should be noted, that Ass(R0) (H-R+(i) (R/I)(n)) 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. [References: 7]