The generating hypothesis in the derived category of \(R\)-modules
type: Item
instance of: scholarly article
title: The generating hypothesis in the derived category of \(R\)-modules
zbMATH Open document ID: 1112.55011
author: Keir Lockridge
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0511534
review text: Freyd's generating hypothesis states that a map between finite spectra that induces the zero map on stable homotopy groups is null-homotopic [\textit{P. Freyd}, Proc. Conf. Categor. Algebra, La Jolla 1965, 121--172 (1966; Zbl 0195.52901)]. In this paper, a version of Freyd's generating hypothesis is considered for a triangulated category \(\mathcal D\) with a distinguished object \(S\) such that the endomorphism ring \(R\) of \(S\) is commutative and concentrated in degree zero. The main theorem is that if the category \(\mathcal D\) admits arbitrary coproducts, then the generating hypothesis is true if and only if \(R\) is von Neumann regular. Here, a commutative ring is von Neumann regular if every principal ideal is generated by an idempotent. An example of such category is the derived category of \(R\)-modules for commutative ring \(R\). The necessity of the theorem follows from two necessary conditions which are equivalent to the condition that \(R\) is von Neumann regular. The sufficiency is proved by showing a property of a von Neumann regular ring \(R\), namely that every finitely generated submodule of a projective \(R\)-module is a summand. Furthermore, similar results are obtained by replacing the finiteness condition, and in a Noetherian stable homotopy category.
reviewed by: Katsumi Shimomura
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 5121389
zbMATH Keywords: generating hypothesis
zbMATH Keywords: derived category of \(R\)-modules
MaRDI profile type: Publication
OpenAlex ID: W2150144085
arXiv ID: math/0511534
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DOI: 10.1016/J.JPAA.2006.01.018
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description: scientific article