Factorial henselian rings with non-factorial completions
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.
scientific article; zbMATH DE number 64214
scientific article