The chromatic tower for \(D(R)\). With an appendix by Marcel Bökstedt
type: Item
instance of: scholarly article
title: The chromatic tower for \(D(R)\). With an appendix by Marcel Bökstedt
zbMATH Open document ID: 0793.18008
DOI: 10.1016/0040-9383(92)90047-L
author: Amnon Neeman
published in: Topology
publication date: 1993-05-16T00:00:00Z
review text: The main aim of the paper is to give a correct proof of a result of \textit{M. J. Hopkins} [Homotopy theory, Proc. Symp. Durham 1985, Lond. Math. Soc. Lect. Note Ser. 117, 73-96 (1987; Zbl 0657.55008)] which asserts that if \(R\) is a commutative ring and \(D^ b(R)\) is the derived category of bounded complexes of finitely generated projective \(R\)-modules then the map \[ f: \begin{pmatrix} \hbox{Triangulated full subcategories of \(D^ b(R)\)}\\ \hbox{closed under direct summands}\end{pmatrix} \to \begin{pmatrix} \hbox{Subsets of \(\text{Spec}(R)\)}\\ \hbox{closed under specializations}\end{pmatrix} \] given by associating to \({\mathcal L}\subseteq D^ b(R)\) the set \(f({\mathcal L})= \{p\in \text{Spec}(R)\mid p\in \text{supp }X\) for some \(X\in{\mathcal L}\}\) is an isomorphism with the inverse \(g\) which associates to \(P\) the smallest triangulated subcateogry of \(D^ b(R)\) closed under specialization and containing \(R/p\) for all \(p\in P\). The original proof of Hopkins contains a gap and the theorem does not hold for an arbitrary ring \(R\). A counterexample is presented in the paper and a correct proof of Hopkin's theorem is given under some additional hypothesis on \(R\), which includes the case \(R\) is Noetherian. An appendix by M. Bökstedt contains a version of Hopkin's theorem for \(R\) arbitrary.
reviewed by: Daniel Simson
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 55Q10
zbMATH DE Number: 166187
zbMATH Keywords: derived category
zbMATH Keywords: Noetherian ring
zbMATH Keywords: prime spectrum
zbMATH Keywords: triangulated category
zbMATH Keywords: stable homotopy theory
zbMATH Keywords: localizing subcategory
MaRDI profile type: Publication
Recommended article: Colocalizing subcategories of D(R)
Recommended article: Classifying subcategories of modules over a commutative noetherian ring
Recommended article: Regular subcategories in bounded derived categories of affine schemes
Recommended article: Triangulated Equivalences Involving Gorenstein Projective Modules
Recommended article: Subcategories of the derived category and cotilting complexes
Recommended article: Liftable derived equivalences and objective categories
Recommended article: ON DERIVED EQUIVALENT COHERENT RINGS
Recommended article: Derived categories of finite spaces and Grothendieck duality
Recommended article: Q4012108
Recommended article: The Brown representability theorem and phantomless triangulated categories
description: scientific article; zbMATH DE number 166187