Finiteness of local homology modules

Rezaei, Shahram

Archivum mathematicum, Tome 56 (2020) no. 1, pp. 31-41

Résumé

Let $I$ be an ideal of Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of weakly colaskerian modules and by using this concept, we give some vanishing and finiteness results for local homology modules. Let $I_{M}:=\operatorname{Ann}_{R}(M/IM)$, we will prove that for any integer $n$ \begin{enumerate} \item[(i)] If $N$ is a weakly colaskerian linearly compact $R$-module such that $(0:_N {I_M})\neq 0$ then $$ \operatorname{width}_{I_M}(N)= \inf\{i\mid \operatorname{H}_i^{I_M}(N)\neq 0 \} =\inf\{i \mid \operatorname{H}_i^I(M,N)\neq 0 \}\,. $$ \item[(ii)] If $(R,\frak{m})$ is a Noetherian local ring and $N$ is an artinian $R$-module then \begin{multline*} \cup_{i}\operatorname{Cos}_R\big(\operatorname{H}_i^{I_M}(N)\big)=\cup_{i}\operatorname{Cos}_R\big(\operatorname{H}_i^I(M,N)\big)=\\ \cup_{i}\operatorname{Cos}_R\big(\operatorname{Tor}_i^R(M/IM,N)\big)\,, \end{multline*} \begin{multline*} \inf\{i \mid \operatorname{H}_i^{I_M}(N) \text{ is not Noetherian $R$-module\,} \}=\\ \inf\{i \mid \operatorname{H}_i^I(M,N) \mbox{\ is not Noetherian $R$-module\,}\}\,. \end{multline*} \end{enumerate}

Détail

DOI : 10.5817/AM2020-1-31

Classification : 13D45, 16E30
Keywords: coregular sequence; local homology; weakly colaskerian