Computing j-multiplicity

TL;DR

This paper investigates the computability of the j-multiplicity invariant in Noetherian local rings, providing new proofs for key formulas and enhancing understanding of its calculation methods.

Contribution

It offers new proofs for the length and additive formulas of j-multiplicity, advancing the computational techniques for this invariant.

Findings

Provided a new proof for the length formula of j-multiplicity

Established an additive formula for j-multiplicity

Enhanced methods for computing j-multiplicity in local rings

Abstract

The j-multiplicity is an invariant that can be defined for any ideal in a Noetherian local ring $(R, m)$. It is equal to the Hilbert-Samuel multiplicity if the ideal is $m$-primary. In this paper we explore the computability of the j-multiplicity, giving another proof for the length formula and the additive formula.