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Some characterizations of relative sequentially Cohen–Macaulay and relative Cohen–Macaulay modules

Department of Engineering Sciences, Faculty of Advanced Technologies, University of Mohaghegh Ardabili Namin, Ardabil, Iran

E-mail Address: zargar9077@gmail.com

E-mail Address: m.zargar@uma.ac.ir

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https://doi.org/10.1142/S0219498825501440Cited by:0 (Source: Crossref)

Abstract

Let $M$ be an $R$ -module over a Noetherian ring $R$ and $𝔞$ an ideal of $R$ with $c = c d (𝔞, R)$ . First, over an $𝔞$ -relative Cohen–Macaulay local ring $(R, 𝔪)$ , we provide a characterization of the $𝔞$ -relative sequentially Cohen–Macaulay modules $M$ in terms of $𝔞$ -relative Cohen–Macaulayness of the $R$ -modules ${E x t}_{R}^{c - i} (M, D_{𝔞})$ for all $i \geq 0$ , where $D_{𝔞} = {H o m}_{R} (H_{𝔞}^{c} (R), E (R / 𝔪))$ . Next, we prove that $M$ is finite $𝔞$ -relative Cohen–Macaulay if and only if $H_{i} (L Λ_{𝔞} (H_{𝔞}^{c d (𝔞, M)} (M))) = 0$ for all $i \neq c d (𝔞, M)$ and $H_{c d (𝔞, M)} (L Λ_{𝔞} (H_{𝔞}^{c d (𝔞, M)} (M))) ≅ {\hat{M}}^{𝔞}$ . Finally, we provide another characterization of $𝔞$ -relative sequentially Cohen–Macaulay modules $M$ in terms of vanishing of the local homology modules $H_{j} (L Λ_{𝔞} (H_{𝔞}^{i} (M))) = 0$ for all $0 \leq i \leq c d (𝔞, M)$ and for all $j \neq i$ .

Communicated by P. Ara

Keywords:

AMSC: 13D45, 13C14, 13D02

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