COFINITENESS AND NON-VANISHING OF LOCAL COHOMOLOGY MODULES

Let R be a commutative Noetherian local ring, I an ideal of R, and let M be a non-zero finitely generated R-module. In this paper, we establish some new properties of the local cohomology modules H-I(i)(M), i >= 0. In particular, we show that if (R, m) is a Noetherian local integral domain of dimension d <= 4 which is a homomorphic image of a Cohen-Macaulay ring and x(1),..., x(n) is a part of a system of parameters for R, then for all i >= 0, the R-modules H-I(i)(R) are I-cofinite, where I = (x(1),..., x(n)). Also, we prove that if (R, m) is a Noetherian local ring of dimension d and x(1),..., x(t) is a part of a system of parameters for R, then H-m(d-t)(H-(x1,...,xt)(t)(R)) not equal 0. In particular, mu(d) (t)(m, H-(x1....,xt)(t) (R)) not equal 0 and injdim(R)(H-(x1,...,xt)(t)(R)) >= d - t.