On natural homomorphisms of local cohomology modules

Let $M$ be a non-zero finitely generated module over a finite dimensional 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 that the natural homomorphisms Tor$^R_c(k,H^c_I(M))\to k\otimes_R M$ and Ext$^{d}_R(k,H^c_I(M))\to {\rm Ext}^t_R(k, M)$ are non-zero where $d:=t-c$. In fact for a Cohen-Macaulay module $M$ we will show that the homomorphism Ext$^d_R(k,H^c_I(M))\to {\rm Ext}^t_R(k, M)$ 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 Tor$^R_c(k,H^c_I(M))\to k\otimes_R M$. Moreover we will construct the natural homomorphism ${\rm Tor}^R_c(k, H^c_I(M))\to {\rm Tor}^R_c(k, H^c_J(M))$ for the ideals $J\subseteq I$ with $c = {\rm grade}(I,M)= {\rm grade}(J,M)$. There are several sufficient conditions on $I$ and $J$ to prove this homomorphism is an isomorphism.