Lengths and multiplicities of integrally closed modules over a two-dimensional regular local ring

TL;DR

This paper extends classical length formulas to integrally closed modules over two-dimensional regular local rings, providing new insights into their structure and multiplicities.

Contribution

It introduces an analogue of the Hoskin-Deligne length formula for integrally closed modules over two-dimensional regular local rings.

Findings

Derived a length formula for integrally closed modules

Obtained a formula for Buchsbaum-Rim multiplicity

Extended classical results to a broader class of modules

Abstract

Let $(R,{\bf m})$ be a two-dimensional regular local ring with infinite residue field. We prove an analogue of the Hoskin-Deligne length formula for a finitely generated, torsion-free, integrally closed $R$-module $M$. As a consequence, we get a formula for the Buchsbaum-Rim multiplicity of $F/M$, where $F = M^{**}$.