Variants of normality for Noetherian schemes

TL;DR

This paper introduces a unified approach to various notions of normality for Noetherian schemes, defining them for pairs of schemes and subschemes, which simplifies proofs and extends applicability.

Contribution

It defines normality variants for pairs of schemes and subschemes, ensuring preservation under completions, thus generalizing and simplifying existing theories.

Findings

Definitions are preserved under completions.

Shortened proofs and more general results.

Unified treatment of normality variants.

Abstract

This note presents a uniform treatment of normality and three of its variants---topological, weak and seminormality---for Noetherian schemes. The key is to define these notions for pairs $(Z, X)$ consisting of a (not necessarily reduced) scheme $X$ and a closed, nowhere dense subscheme $Z$. An advantage of the new definitions is that, unlike the usual absolute ones, they are preserved by completions. This shortens some of the proofs and leads to more general results. Version 2: small changes.