Fitting Ideals without a Presentation
Ayse Sharland, Jacob Smith
TL;DR
This work develops methods to compute higher Fitting ideals $\mathrm{Fitt}_i(f_*\mathcal{O}_{X,0})$ for finite holomorphic map-germs $f:(X,0)\to(\mathbb{C}^{n+1},0)$ without relying on full presentations. It unifies conductor- and Jacobian-based viewpoints to characterize $\mathrm{Fitt}_1$ and its geometric meaning as the ramification/normalization data of the image, including a target-version of Piene’s relation for stable maps. The authors prove an iterative quotient formula $\big(\mathrm{Fitt}_{k+1}^2(f_*\mathcal{O}_{X,0}) : \mathrm{Fitt}_k(f_*\mathcal{O}_{X,0})\big)=\mathrm{Fitt}_{k+2}(f_*\mathcal{O}_{X,0})$ for corank-1, $\mathcal{A}$-finite map-germs with a presentation, and outline an inductive approach via the double point space $D^2(f)$ to extend these results. They also demonstrate practical computations and algorithmic strategies in Singular and Macaulay2, and discuss how these constructions illuminate the structure of multiple point spaces $M_k(f)$ and the associated analytic geometry. The results advance computational techniques for singularity theory by connecting classical invariants to new, presentation-free reformulations.
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.