The p-radical closure of local noetherian rings

TL;DR

This paper introduces the p-radical closure of local noetherian rings, exploring its properties and applications in constructing examples of rings with complex singularity behaviors.

Contribution

It defines the p-radical closure and analyzes its properties, providing new examples of rings with non-finite normalization and non-reduced formal completions.

Findings

Finitely generated intermediate rings have peculiar properties.

Examples of rings with non-finite normalization are constructed.

Rings without resolution of singularities are identified.

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.