The p-radical closure of local noetherian rings

Abstract

Given a local noetherian ring $R$ whose formal completion is integral, we introduce and study the $p$-radical closure $R^{\text{prc}}$. Roughly speaking, this is the largest purely inseparable $R$-subalgebra inside the formal completion $\hat{R}$. It turns out that the finitely generated intermediate rings $R\subset A\subset R^{\text{prc}}$ have rather peculiar properties. They can be used in a systematic way to provide examples of integral local rings whose normalization is non-finite, that do not admit a resolution of singularities, and whose formal completion is non-reduced.

arXiv comment19 pages, change of title, corrected an erroneous application of MacLane's separability criterion, otherwise minor changes. To appear in J. Commut. Algebra