Relatively Noetherian rings, localization and sheaves. II: Structure sheaves
type: Item
instance of: scholarly article
title: Relatively Noetherian rings, localization and sheaves. II: Structure sheaves
zbMATH Open document ID: 0834.16033
DOI: 10.1007/BF00961454
author: C. Vidal Martin
author: Alain Verschoren
published in: \(K\)-Theory
publication date: 1994-07-12T00:00:00Z
review text: As an application of the results obtained in the first part of the paper [see the preceding review Zbl 0834.16032], the authors now show how the relative version of the strong second layer condition allows us to endow \(K(\lambda)\), the generically closed subset of \(\text{Spec}(R)\) canonically associated to a radical \(\lambda\) in \(R\)-mod, with structure sheaves, which generalize most previously studied sheaf constructions in noncommutative algebra. First they develop some general machinery to construct structure sheaves through localization in a rather generic way. The sheaf construction itself, both in the symmetric and the classical case, depends upon the choice of a suitable topology on \(K(\lambda)\). In the classical case, they provide some extra background on localization at relative Ore sets, generalizing and strengthening the results obtained in the absolute, Noetherian case. Then the results of the first part of the paper allow us to associate to any \(R\)-bimodule \(M\) both a symmetric and a classical sheaf of \(R\)- modules on \(K(\lambda)\), whose global sections reduce to the localization of \(M\) at \(\lambda\). This generalizes most previously studied symmetric and nonsymmetric sheaf constructions for modules over noncommutative rings.
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 590952
zbMATH Keywords: strong second layer condition
zbMATH Keywords: structure sheaves
zbMATH Keywords: sheaf constructions in noncommutative algebra
zbMATH Keywords: localizations
zbMATH Keywords: relative Ore sets
zbMATH Keywords: global sections
reviewed by: Yoshiki Kurata
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 590952