Noetherian domains inside a homomorphic image of a completion
Given a field \(k\) and and a field \(L\) of finite type over \(k\), a general problem is to describe the local Noetherian domains \(A\) between \(k\) and \(L\), such that \(A\) has fraction field \(L\) and residue field \(k\). It could be shown that every \(A\) is realizable as the intersection of \(L\) with a homomorphic image of the \(\mathfrak{m}\)-adic completion of a local Noetherian domain (of maximal ideal \(\mathfrak{m}\)) essentially of finite type over \(k\) and having \(L\) as fraction field. However it is not true in general that such intersection is Noetherian. In the paper under review, motivated by the above problem, the authors give the following criterion for the intersection domain to be Noetherian: Let \(R\) be a Noetherian integral domain with fraction field \(L\). Let \(x\) be a nonzero nonunit of \(R\) and let \(R^*\) denote the \((x)\)-adic completion of \(R\). Suppose \(I\) is an ideal of \(R^*\) with the property that \(P \cap R=(0)\) for each \(P\in \text{Ass}(R^*/I)\). Then \(R\to (R^*/I)[1/x]\) is flat if and only if \(L\cap (R^*/I)\) is Noetherian and is realizable as a localization of a subring of \(R[1/x]\). The authors also give several examples where flatness holds, and study some regularity properties related to the above setting.
scientific article; zbMATH DE number 1309132
scientific article