arXiv 1502.011082015-02math.AC

关于广义完备同调模

On generalized completion homology modules

Waqas Mahmood

备注：: 17 pages

摘要：

设$I$是交换Noether环$R$的理想.设$M$和$N$是任何$R$ -模块。我们定义了$i\in \mathbb{Z}$的广义完备同调模$L_iΛ^I (N,M)$作为复体$\lim\limits_{\longleftarrow}(N/I^sN\otimes_R F_{\cdot}^R)$的同调.这里，$F_{\cdot}^R$表示$M$的平坦分辨率。在这篇文章中，我们将证明的消失和非零性质的$L_iΛ^I (N,M)$。我们表示为$H^{i}_{I}(N,M)$（分别为$U^I_i(N,M)$）的广义局部上同调模（分别为.广义局部同调模）。作为一种技术工具，我们将构造$L_iΛ^I (N,M)$、$H^{i}_{I}(N,M)$和$U^I_i(N,M)$的几个自然同态。我们将研究这些自然同态何时是同构。如果$M$是Artinian的且$N$是逆生成的，则证明了对于每个$i\in \mathbb{Z}$，$L_iΛ^I (N,M)$同构于$U^I_i(N,M)$.对于$H^i_{I}(N,M)$也得到了类似的结果。此外，如果$M$和$N$都是用$c=\grade(I,M)$并行生成的。然后我们能够证明几个充分必要条件，使$H^i_{I}(M)=0$对所有$i\neq c.$这里$H^i_{I}(M)$表示普通局部上同调模。

英文摘要：

Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be any $R$-modules. We define the generalized completion homology modules $L_iΛ^I (N,M)$, for $i\in \mathbb{Z}$, as the homologies of the complex $\lim\limits_{\longleftarrow}(N/I^sN\otimes_R F_{\cdot}^R)$. Here $F_{\cdot}^R$ denote a flat resolution of $M$. In this article we will prove the vanishing and non-vanishing properties of $L_iΛ^I (N,M)$. We denote $H^{i}_{I}(N,M)$ (resp. $U^I_i(N,M)$) by the generalized local cohomology modules (resp. the generalized local homology modules). As a technical tool we will construct several natural homomorphisms of $L_iΛ^I (N,M)$, $H^{i}_{I}(N,M)$ and $U^I_i(N,M)$. We will investigate when these natural homomorphisms are isomorphisms. Moreover if $M$ is Artinian and $N$ is finitely generated then it is proven that $L_iΛ^I (N,M)$ is isomorphic to $U^I_i(N,M)$ for each $i\in \mathbb{Z}$. The similar result is obtained for $H^i_{I}(N,M)$. Furthermore if both $M$ and $N$ are finitely generated with $c=\grade(I,M)$. Then we are able to prove several necessary and sufficient conditions such that $H^i_{I}(M)=0$ for all $i\neq c.$ Here $H^i_{I}(M)$ denote the ordinary local cohomology module.

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：

相关论文