Sheaves of abelian \(l\)-groups
type: Item
instance of: scholarly article
title: Sheaves of abelian \(l\)-groups
zbMATH Open document ID: 1293.13002
author: Niels Schwartz
published in: Order
publication date: 2013-06-26T00:00:00Z
review text: Let \(G\) denote an abelian \(l\)-group (lattice ordered group) and let \(R\) denote an integral domain with quotient field \(K.\) A GCD domain \(R\) (every pair of nonzero elements has a GCD) has the property that its group of divisibility \(G(R)=\{uR:u\in K\backslash \{0\}\}\) ordered by \(uR\geq vR\) if and only if \(uR\subseteq vR\) is an \(l\)-group, with \(uR\vee vR=uR\cap vR\) and \(uR\wedge vR=((uR+vR)^{-1})^{-1}\) where, for a fractional ideal \(A,\) we have \(A^{-1}=\{x\in K:\) \(xA\subseteq R\}.\) A Bézout domain (every two generated ideal is principal) is a GCD domain for which \(uR\wedge vR\) gets a more direct description \(uR\wedge vR=uR+vR\) because if \(A=uR\) then \(A^{-1}=\frac{1 }{u}R\) and \((A^{-1})^{-1}=uR.\) So the link between Bézout domains and \(l\) -groups is more direct. Perhaps that is why \textit{P. Jaffard} [J. Math. Pures Appl. (9) 32, 203--280 (1953; Zbl 0051.01303)] and \textit{J. Ohm} [Trans. Am. Math. Soc. 122, 321--333 (1966; Zbl 0142.28502)] were able to show that every \(l\)-group can be realized as the group of divisibility \(G(R)\) of a Bézout domain \(R\), using an easy group algebra construction. \textit{W. Rump} and \textit{Y. C. Yang} [Bull. Lond. Math. Soc. 40, No. 2, 263--273 (2008; Zbl 1151.13001)] studied the correspondence between abelian \(l\)-groups and Bézout domains. They verified Anderson's conjecture (every \(l\)-embedding \(G(R)\rightarrow H\) comes from an extension \(R\rightarrow S\) of Bézout domains) and used the refinement provided by it to extend the Jaffard Ohm correspondence to the correspondence between sheaves and spectral spaces, using \textit{M. Hochster}'s work [Trans. Am. Math. Soc. 142, 43--60 (1969; Zbl 0184.29401)]. They also show that the spectrum of a Bézout domain and the spectrum of its corresponding abelian \(l\)-group present a perfect example for Hochster's duality of spectral spaces. The notion of the spectrum of an abelian \(l\)-group and the corresponding sheaf was introduced by \te…
reviewed by: Muhammad Zafrullah
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 13A05
zbMATH DE Number: 6181431
zbMATH Keywords: lattice-ordered group
zbMATH Keywords: \(l\)-ideal
zbMATH Keywords: prime \(l\)-ideal
zbMATH Keywords: spectrum
zbMATH Keywords: sheaf
zbMATH Keywords: group of divisibility
zbMATH Keywords: Prüfer domain
zbMATH Keywords: group of Cartier divisors
zbMATH Keywords: valuation
MaRDI profile type: Publication
full work available at URL: s11083-012-9258-0
OpenAlex ID: W1980765583
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DOI: 10.1007/S11083-012-9258-0
description: scientific article