Fitting Ideals without a Presentation

TL;DR

This paper explores new methods to compute Fitting ideals of pushforward modules for finite holomorphic map-germs, extending classical results and providing iterative and quotient-based formulas for various coranks.

Contribution

It introduces alternative constructions for Fitting ideals, generalizes existing formulas for corank 1, and relates the first Fitting ideal to Jacobian and ramification ideals for stable map-germs.

Findings

Iterative calculation of Fitting ideals for corank 1 map-germs.

Expression of the first Fitting ideal as a quotient of Jacobian and ramification ideals.

Extension of classical results to broader classes of map-germs.

Abstract

In this article, we investigate alternative construction of Fitting ideals of pushforward modules $f_*\mathcal{O}_{X,0}$ for finite and holomorphic map-germs from an $n$-dimensional Cohen-Macaulay space $(X,0)$ to $(\mathbb{C}^{n+1},0)$. For corank 1 map-germs, we generalize a result of D. Mond and R. Pellikaan to iteratively calculate $k$-th Fitting ideals as ideal quotients of lower ones. We also show that for a stable map-germ of any corank, the first Fitting ideal can be calculated as a quotient ideal of the Jacobian of the image and the pushforward of the ramification ideal, which is a modification of classical result of due to Piene.