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On natural homomorphisms of local cohomology modules (1406.7461v1)

Published 29 Jun 2014 in math.AC

Abstract: 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$R_c(k,Hc_I(M))\to k\otimes_R M$ and Ext${d}_R(k,Hc_I(M))\to {\rm Ext}t_R(k, M)$ are non-zero where $d:=t-c$. In fact for a Cohen-Macaulay module $M$ we will show that the homomorphism Ext$d_R(k,Hc_I(M))\to {\rm Ext}t_R(k, M)$ is injective (resp. surjective) if and only if the homomorphism $H{d}{\mathfrak{m}}(Hc_{I}(M))\to Ht_{\mathfrak{m}}(M)$ is injective (resp. surjective) under the additional assumption of vanishing of Ext modules. The similar results are obtained for the homomorphism Tor$R_c(k,Hc_I(M))\to k\otimes_R M$. Moreover we will construct the natural homomorphism ${\rm Tor}R_c(k, Hc_I(M))\to {\rm Tor}R_c(k, Hc_J(M))$ 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.

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