ON NATURAL HOMOMORPHISMS OF LOCAL COHOMOLOGY MODULES

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, m) with dim(R)(M) = t. Let I be an ideal of R with grade(I, M) = c. In this article we will investigate several natural homomorphisms of local cohomology modules. The main purpose of this article is to investigate when the natural homomorphisms gamma : Tor(c)(R)(k, H-I(c)(M)) -> k circle times(R) M and eta : Ext(R)(d)(k, H-I(c)(M)) -> Ext(R)(t)(k, M) are non-zero where d := t-c. In fact for a Cohen-Macaulay module M we will show that the homomorphism eta is injective (resp. surjective) if and only if the homomorphism H-m(d)(H-I(c)(M)) -> H-m(t)(M) is injective (resp. surjective) under the additional assumption of vanishing of Ext modules. The similar results are obtained for the homomorphism gamma. Moreover we will construct the natural homomorphism Tor(c)(R)(k, H-I(c)(M)) -> Tor(c)(R)(k, H-J(c)(M)) for the ideals J subset of I with c = grade(I, M) = grade(J, M). There are several sufficient conditions on I and J to provide this homomorphism is an isomorphism.