Discrete valuation domains and ranks of their maximal extensions
type: Item
instance of: scholarly article
title: Discrete valuation domains and ranks of their maximal extensions
zbMATH Open document ID: 0608.13010
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108016
review text: Let \(R\) denote a valuation domain, \(\overline R\) its completion and \(S\) a maximal immediate extension of \(R\). Let \(\mathrm{Spec}(R)\) be the set of all prime ideals of \(R\), totally ordered by inclusion. Let \(P\in \mathrm{Spec}(R)\). The completion defect \(c_R(P)\) at \(P\) is defined as the rank of the torsionfree \(R/P\)-module \(\overline{R/P}\). The total defect \(d_R(P)\) at \(P\) is the rank of the torsion free \(R/P\)-module \(S/PS\). Some formulae are proved that connect the functions \(c_R\) and \(d_R\), e.g.: Let \(R\) be a discrete valuation domain and \(P\) a prime nonmaximal ideal of \(R\). Then \(d_R(P) = c_R(P) \sup \{d_R(Q); Q\supset P, Q\in \mathrm{Spec}(R)\}\). Finally, the authors completely solve the problem of determining the function ``completion defect'' that can arise from discrete valuation domains \(R\) of non-zero characteristic satisfying the a.c.c. on prime ideals.
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13B02
zbMATH DE Number: 3983342
zbMATH Keywords: maximal immediate extension
zbMATH Keywords: completion defect
zbMATH Keywords: total defect
author: Alberto Facchini
author: P. Zanardo
reviewed by: Štefan Porubský
MaRDI profile type: Publication
cites work: Q3413653
cites work: Q5661989
cites work: Injective Modules Over Prufer Rings
cites work: Q4052239
cites work: Q5611879
cites work: Q5735274
cites work: On the completion of a valuation ring
cites work: Some cardinal invariants for valuation domains
cites work: Q5656860
cites work: Valuation domains without pathological modules
cites work: Residual fields in valuation theory.
description: scientific article; zbMATH DE number 3983342