On the colength of fractional ideals

TL;DR

This paper develops a recursive formula for calculating the colength of fractional ideals in complete admissible rings, extending previous results and providing explicit formulas for rings with up to three minimal primes.

Contribution

It introduces a recursive approach based on the minimal primes and value set maximal points, generalizing prior work to broader classes of rings.

Findings

Recursive formula for colength in terms of minimal primes

Closed formulas for rings with two or three minimal primes

Improves upon previous results by Barucci, D'Anna, and Fröberg

Abstract

The main result in this paper is to supply a recursive formula, on the number of minimal primes, for the colength of a fractional ideal in terms of the maximal points of the value set of the ideal itself. The fractional ideals are taken in the class of complete admissible rings, a more general class of rings than those of algebroid curves. For such rings with two or three minimal primes, a closed formula for that colength is provided, so improving results by Barucci, D'Anna and Fr\"oberg.