Variants of normality and steadfastness deform

TL;DR

This paper investigates how properties like $p$-seminormality and steadfastness behave under deformations and extensions in rings, providing new insights and proofs related to the cancellation problem and normality conditions.

Contribution

It proves that $p$-seminormality and steadfastness deform in reduced Noetherian local rings and are stable under adjoining formal power series variables, offering new proofs for normality deformation.

Findings

$p$-seminormality and steadfastness deform in reduced Noetherian local rings.

These properties are stable under adjoining formal power series variables.

New proofs for deformation of normality and weak normality are provided.

Abstract

The cancellation problem asks whether $A[X_1,X_2,\ldots,X_n] \cong B[Y_1,Y_2,\ldots,Y_n]$ implies $A \cong B$. Hamann introduced the class of steadfast rings as the rings for which a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer holds. By work of Asanuma, Hamann, and Swan, steadfastness can be characterized in terms of $p$-seminormality, which is a variant of normality introduced by Swan. We prove that $p$-seminormality and steadfastness deform for reduced Noetherian local rings. We also prove that $p$-seminormality and steadfastness are stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. Our methods also give new proofs of the facts that normality and weak normality deform, which are of independent interest.