arXiv 1406.74612014-06-29math.AC

关于局部上同调模的自然同态

On natural homomorphisms of local cohomology modules

Waqas Mahmood

备注：: submitted, keyword: Commutative Algebra

摘要：

设$M$是有限维交换Noether局部环$(R,\mathfrak{m})$上的非零生成模，其dim为$_R(M)=t$ .设$I$是$R$的理想，具有等级$(I,M)=c$。本文主要研究局部上同调模的几种自然同态。本文的主要目的是研究自然同态Tor $^R_c(k,H^c_I(M))\to k\otimes_R M$和Ext $^{d}_R(k,H^c_I(M))\to {\rm Ext}^t_R(k, M)$是非零的，其中$d:=t-c$ .事实上，对于Cohen-Macaulay模$M$，我们将证明同态Ext $^d_R(k,H^c_I(M))\to {\rm Ext}^t_R(k, M)$是单射的（分别是）。满射）当且仅当同态$H^{d}_{\mathfrak{m}}(H^c_{I}(M))\to H^t_{\mathfrak{m}}(M)$是单射（分别为满射）的额外假设下消失的Ext模块。对同态Tor $^R_c(k,H^c_I(M))\to k\otimes_R M$也得到了类似的结果.此外，我们将构造理想$J\subseteq I$与$c = {\rm grade}(I,M)= {\rm grade}(J,M)$的自然同态${\rm Tor}^R_c(k, H^c_I(M))\to {\rm Tor}^R_c(k, H^c_J(M))$。在$I$和$J$上有几个充分条件来证明这个同态是同构。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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