New characterizations of restricted injective dimensions for complexes
type: Item
instance of: scholarly article
title: New characterizations of restricted injective dimensions for complexes
zbMATH Open document ID: 1461.13014
author: Zhuang Miao
author: Dejun Wu
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2021-05-05T00:00:00Z
review text: Let $R$ be a commutative noetherian ring, and $\mathcal{D}(R)$ denote the derived category of $R$. For any $X\in \mathcal{D}_{\sqsubset}(R)$, the small restricted injective dimension of $X$ is defined as \[ \operatorname{rid}_{R}(X) = \sup \Big\{- \inf \left(\mathbf{R}\mathrm{Hom}_{R}(M,X)\right): M \text{ is a finitely generated } R\text{-module of finite projective dimension} \Big\}, \] and for any $X\in \mathcal{D}_{\sqsupset}(R)$, the small restricted flat dimension of $X$ is defined as \[ \operatorname{rfd}_{R}(X) = \sup \Big\{\sup \left(M\otimes_{R}^{\mathbf L} X\right): M \text{ is a finitely generated } R\text{-module of finite projective dimension} \Big\}. \] The authors of this paper give a new characterization of the small restricted injective dimension in their main theorem, and then derive two base change inequalities as a corollary to it. As an application, they further provide a characterization of almost Cohen-Macaulay rings as follows: Theorem. Let $R$ be a local ring. Then the following assertions are equivalent: \begin{itemize} \item[(i)] $R$ is almost Cohen-Macaulay, i.e. $\operatorname{cmd}(R):= \dim(R)- \operatorname{depth}(R) \leq 1$. \item[(ii)] $\operatorname{rfd}_{R}(X)= \operatorname{fd}_{R}(X)$ for every $X\in D_{\square}(R)$ of finite flat dimension. \item[(iii)] $\operatorname{rfd}_{R}(M)= \operatorname{fd}_{R}(M)$ for every $R$-module of finite flat dimension. \end{itemize}
reviewed by: Hossein Faridian
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 18G35
zbMATH DE Number: 7344143
zbMATH Keywords: restricted injective dimension
zbMATH Keywords: Bass formula
zbMATH Keywords: Cohen-Macaulay ring
Wikidata QID: Q114220150
MaRDI profile type: Publication
full work available at URL: s13226-020-0484-7
OpenAlex ID: W3119490287
cites work: Homological dimensions of unbounded complexes
cites work: Q3843140
cites work: Gorenstein dimensions
cites work: Restricted homological dimensions and Cohen-Macaulayness
cites work: On Gorenstein projective, injective and flat dimensions -- a functorial description with applications
cites work: Ascent Properties of Auslander Categories
cites work: Q4451109
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: On the vanishing of \(\lim_{(i)}\)
cites work: ALMOST COHEN–MACAULAY MODULES
cites work: Gorenstein Injective and Gorenstein Flat Dimensions Under Base Change
cites work: Transfer of Gorenstein Dimensions of Unbounded Complexes Along Ring Homomorphisms
cites work: Depth formulas, restricted Tor-dimension under base change
cites work: Gorenstein dimensions of unbounded complexes under base change
cites work: On Restricted Injective Dimensions of Complexes
cites work: Width of complexes of modules
DOI: 10.1007/S13226-020-0484-7
description: scientific article; zbMATH DE number 7344143