Colocalization of local homology modules
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Let ${I}$ be an ideal of Noetherian local ring $(R,\fm)$ and $M$ an artinian $R$-module. In this paper, we study colocalization of local homology modules. In fact we give Colocal-global Principle for the artinianness and minimaxness of local homology modules, which is a dual case of Local-global Principle for the finiteness of local cohomology modules. We define the representation dimension $ r^{I}(M)$ of $M$ and the artinianness dimension $ a^{I}(M)$ of $M$ relative to $I$ by $ r^{I}(M)=\inf\{i\in\mathbb{N}_{0}:\HE_i^{I}(M)$ is not representable$\}$, and $a^I (M)=\inf\{i\in\mathbb{N}_{0}:\HE_i^{I}(M)$ is not artinian$\}$ and we will prove that \begin{itemize} \item[i)] $ a^I (M)=r^{I}(M)=\inf\{ r^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R) \}\geq \inf\{ a^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R) \}$, \item[ii)] $\inf\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not minimax$\rbrace=\inf\{ r^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R)\setminus\lbrace\fm\rbrace \}$. \end{itemize} Also, we define the upper representation dimension $ R^{I}(M)$ of $M$ relative to $I$ by $R^{I}(M)=\sup\{i\in\mathbb{N}_{0}: \HE_i^{I}(M)$ is not representable$\}$, and we will show that \begin{itemize} \item[i)] $\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M) \neq 0 \rbrace=\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not artinian$\rbrace=\sup\{ R^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R)\}$, \item[ii)] $\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not finitely generated$\rbrace=\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not minimax$\rbrace=\sup\{ R^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R)\setminus\lbrace\fm\rbrace \}$. \end{itemize}
Abstract
Let ${I}$ be an ideal of Noetherian local ring $(R,\fm)$ and $M$ an artinian $R$-module. In this paper, we study colocalization of local homology modules. In fact we give Colocal-global Principle for the artinianness and minimaxness of local homology modules, which is a dual case of Local-global Principle for the finiteness of local cohomology modules. We define the representation dimension $ r^{I}(M)$ of $M$ and the artinianness dimension $ a^{I}(M)$ of $M$ relative to $I$ by $ r^{I}(M)=\inf\{i\in\mathbb{N}_{0}:\HE_i^{I}(M)$ is not representable$\}$, and $a^I (M)=\inf\{i\in\mathbb{N}_{0}:\HE_i^{I}(M)$ is not artinian$\}$ and we will prove that \begin{itemize} \item[i)] $ a^I (M)=r^{I}(M)=\inf\{ r^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R) \}\geq \inf\{ a^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R) \}$, \item[ii)] $\inf\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not minimax$\rbrace=\inf\{ r^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R)\setminus\lbrace\fm\rbrace \}$. \end{itemize} Also, we define the upper representation dimension $ R^{I}(M)$ of $M$ relative to $I$ by $R^{I}(M)=\sup\{i\in\mathbb{N}_{0}: \HE_i^{I}(M)$ is not representable$\}$, and we will show that \begin{itemize} \item[i)] $\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M) \neq 0 \rbrace=\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not artinian$\rbrace=\sup\{ R^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R)\}$, \item[ii)] $\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not finitely generated$\rbrace=\sup\{i\in\mathbb{N}_{0}: \HE_{i}^{I}(M)$ is not minimax$\rbrace=\sup\{ R^{IR_{\fp}}(_{\fp}M) : \fp \in \Spec(R)\setminus\lbrace\fm\rbrace \}$. \end{itemize}
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