Factorial henselian rings with non-factorial completions
type: Item
instance of: scholarly article
title: Factorial henselian rings with non-factorial completions
zbMATH Open document ID: 0771.13002
DOI: 10.1016/0022-4049(92)90137-5
author: Dana Weston
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: This paper gives a procedure for constructing henselian rings which are factorial while their completions are not. Thus one gets a noetherian, local analytically normal domain \(A\) such that the class group of \(A\) and that of its henselization \(A^ h\) are trivial while the completion \(\hat A\) of \(A\) with respect to the topology given by the maximal ideal, has a nontrivial class group. What is more, the class group of \(\hat A\) can be chosen arbitrarily large. -- The construction also shows that (1) the henselization \(A^ h\) is a non-approximation ring (in a characteristic free situation), (2) there exists an analytically normal local domain \(a\) of Krull dimension \(>2\) with class group of \(A\), a torsion group such that for each nonprincipal divisorial ideal \({\mathfrak A}\) of \(A\), no \(q\)-fold sum of \({\mathfrak A}\) descends to an \(A\)-module, and (3) there exists a Cohen-Macaulay local ring \(a\) with all formal fibres Gorenstein, but with non Gorenstein module.
Mathematics Subject Classification ID: 13B35
Mathematics Subject Classification ID: 13B40
Mathematics Subject Classification ID: 13J15
Mathematics Subject Classification ID: 13F15
zbMATH DE Number: 64214
zbMATH Keywords: constructing henselian rings
zbMATH Keywords: completion
zbMATH Keywords: nontrivial class group
reviewed by: N. Sankaran
MaRDI profile type: Publication
full work available at URL: 0022-4049(92)90137-5
OpenAlex ID: W2078988523
cites work: The Brauer group of a ringed space
cites work: The Brauer Group of a Commutative Ring
cites work: On Maximally Central Algebras
cites work: Q3953883
cites work: Rationale Singularitäten komplexer Flächen
cites work: Local domains with bad sets of formal prime divisors
cites work: Q4116543
cites work: Local Cohomology Modules for Normal Domains
cites work: Die Sätze von Bertini für lokale Ringe
cites work: Q5670698
cites work: Minimal injective resolutions with applications to dualizing modules and Gorenstein modules
cites work: The divisor class group of a certain Krull domain
cites work: A complex associated with a symmetric matrix
cites work: Q5605293
cites work: The Brauer group of \(A[T]\)
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. (Séconde partie)
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: Der kanonische Modul eines Cohen-Macaulay-Rings. (The canonical moduls of a Cohen-Macaulay-ring)
cites work: Some applications of the Frobenius in characteristic 0
cites work: Cohen-Macaulay Rings and Ideal Theory in Rings of Invariants of Algebraic Groups
cites work: Rational singularities, with applications to algebraic surfaces and unique factorization
cites work: Q3739243
cites work: A note on factorial rings
cites work: Q3941519
cites work: The Converse to a Theorem of Sharp on Gorenstein Modules
cites work: Universell japanische Ringe mit night offenem regulärem ort
cites work: Q5524437
cites work: On unique factorization domains
cites work: Gorenstein modules
cites work: On descent in dimension two and non-split Gorenstein modules
cites work: Generating absolute prime ideals
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description: scientific article; zbMATH DE number 64214