On the perturbations of Noetherian local domains

TL;DR

This paper investigates how certain algebraic properties of Noetherian local rings, such as being reduced, integral domain, or normal, are affected by small perturbations of their defining equations, identifying specific cases where stability holds.

Contribution

It establishes conditions under which properties like being an integral domain, normal, or reduced are stable under small perturbations in Noetherian local rings.

Findings

Stability of integral domain under perturbations in factorial excellent Henselian local rings.

Normality remains stable under perturbations in excellent local complete intersections with characteristic zero.

Reducedness is stable under perturbations in certain complete intersections and factorial Nagata local rings.

Abstract

We study how the properties of being reduced, integral domain, and normal, behave under small perturbations of the defining equations of a noetherian local ring. It is not hard to show that the property of being a local integral domain (reduced, normal ring) is not stable under small perturbations in general. We prove that perturbation stability holds in the following situations: (1) perturbation of being an integral domain for factorial excellent Henselian local rings; (2) perturbation of normality for excellent local complete intersections containing a field of characteristic zero; and (3) perturbation of reducedness for excellent local complete intersections containing a field of characteristic zero, and for factorial Nagata local rings.