Variants of normality and steadfastness deform

TL;DR

The paper investigates how variants of normality, specifically $p$-seminormality, and steadfastness interact with deformation and base-change. By extending Heitmann’s deformation strategy to the $p$-seminormal setting, it establishes that $p$-seminormality and steadfastness deform for reduced Noetherian local rings and remain stable under adjoining formal power series variables for reduced (not necessarily Noetherian) rings. It also provides new proofs that normality and weak normality deform, enriching the understanding of these properties under Cartier-type deformations. The work connects cancellation problems to a robust deformation theory, enabling construction and analysis of steadfast rings in broader contexts.

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.

Figures (1)

• Figure 1: Variants of normality and steadfastness.

Theorems & Definitions (44)

• Theorem \ref{thm:pseminormallifts}
• Theorem \ref{thm:pseminormalpowerseries}
• Theorem \ref{thm:nwnlift}
• Definition 2.1: Ham75
• Definition 2.2: Swa80 p. 212; Yan85 p. 89 and p. 91
• Definition 2.3: Swa80 Definition on p. 210; Yan85 p. 94
• Theorem 2.4: Swa80 p. 216; Yan85 Remark 1
• proof
• Theorem 2.5: Swa80
• Definition 2.6: Swa80