$\mathfrak{m}$-adic Perturbations in Noetherian Local Rings

TL;DR

This paper introduces new methods to analyze the stability of algebraic invariants like Hilbert-Samuel and Hilbert-Kunz multiplicities under small perturbations in Noetherian local rings, enhancing understanding of their behavior.

Contribution

It develops novel techniques for studying $rak{m}$-adic stability and applies them to understand the behavior of key multiplicities under perturbations in Noetherian local rings.

Findings

Proves stability results for Hilbert-Samuel multiplicities.

Establishes behavior of Hilbert-Kunz multiplicities under perturbations.

Provides new tools for local algebra analysis.

Abstract

We develop new methods to study $\mathfrak{m}$-adic stability in an arbitrary Noetherian local ring. These techniques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine $\mathfrak{m}$-adic perturbations.