Injective modules under faithfully flat ring extensions
type: Item
instance of: scholarly article
title: Injective modules under faithfully flat ring extensions
zbMATH Open document ID: 1361.13006
author: Lars Winther Christensen
author: Fatih Köksal
published in: Proceedings of the American Mathematical Society
publication date: 2016-03-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1406.7791
Mathematics Subject Classification ID: 13C11
Mathematics Subject Classification ID: 13B40
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 6549104
zbMATH Keywords: injective module
zbMATH Keywords: faithfully fat
zbMATH Keywords: ring extension
zbMATH Keywords: dimension
zbMATH Keywords: exact sequence
MaRDI profile type: Publication
OpenAlex ID: W2964300440
Wikidata QID: Q114094239
arXiv ID: 1406.7791
cites work: Q4545926
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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: Q4296949
DOI: 10.1090/PROC/12791
review text: Let \(\mathbf{R}\text{Hom}_R(-, N)\) and \(\text{id}_R(N)\) denote the right derived homomorphism functor and the injective dimension of \(N\), respectively. Also, assume that \(S\) is a flat \(R\)-algebra in which \(R\) is a Noetherian ring. In the paper under review, the authors prove that if every flat module has finite projective dimension, then NEWLINE\[NEWLINE\text{id}_R(N)\geq \text{id}_S \mathbf{R}\text{Hom}_R(S, N)\geq \text{id}_R \mathbf{R}\text{Hom}_R(S, N);NEWLINE\]NEWLINE moreover, equalities hold if \(S\) is a faithfully flat \(R\)-algebra. So, in particular, the injectivity of \(\text{Hom}_R(S, N)\) and the vanishing of the functor \(\text{Ext}_R^n(-,N)\) on \(S\), for every \(n\geq 1\) implies the injectivity of \(N\).
reviewed by: F. Shaveisi
description: scientific article; zbMATH DE number 6549104