Abstract :
Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R,m) with dimR (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 γ:TorRc(k,HcI(M))→k⊗RM and η:ExtdR(k,HcI(M))→ExttR(k,M) are non-zero where d:=t−c. In fact for a Cohen-Macaulay module M we will show that the homomorphism η is injective (resp. surjective) if and only if the homomorphism Hdm(HcI(M))→Htm(M) is injective (resp. surjective) under the additional assumption of vanishing of Ext modules. The similar results are obtained for the homomorphism γ. Moreover we will construct the natural homomorphism TorRc(k,HcI(M))→TorRc (k,HcJ(M)) for the ideals J⊆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.
