Abstract :
Let $M$ be a non-zero finitely generated module over a commutative Noetherian local ring $(R,mathfrak{m})$ 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 $gamma: Tor^{R}_c(k,H^c_I(M))to kotimes_R M$ and $eta: Ext^{d}_R(k,H^c_I(M))to Ext^{t}_R(k, M)$ 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 $H^{d}_{mathfrak{m}}(H^c_{I}(M))to H^t_{mathfrak{m}}(M)$ 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 $Tor^{R}_c(k, H^c_I(M))to Tor^{R}_c(k, H^c_J(M))$ for the ideals $Jsubseteq I$ with $c = grade(I,M)= grade(J,M)$. There are several sufficient conditions on $I$ and $J$ to provide this homomorphism is an isomorphism.
