FITTING IDEALS WITHOUT A PRESENTATION

Author

Jacob L. Smith, University of Rhode IslandFollow

Date of Award

2024

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics and Applied Mathematical Sciences

First Advisor

Thomas J. Sharland

Abstract

In this thesis, we are interested in finding alternative methods for calculating Fitting ideals that appear in the study of the multiple point spaces in the target of finite holomorphic map-germs. If f:(ℂⁿ, 0) -> (ℂⁿ⁺¹, 0) is a finite holomorphic map-germ, then the ring O_n is a finite O_n+1-module. Moreover, O_n has a presentation given by a square and symmetric matrix over O_n+1. In a classic paper of Mond and Pellikaan, it was shown that the Fitting ideals of the associated presentation matrix define the multiple point spaces of f in the target and endow them with an appropriate analytic structure. In a classic paper of Mond and Pellikaan, it was shown that the Fitting ideals of the associated presentation matrix define the multiple point spaces of f in the target and endow them with an appropriate analytic structure. In the same paper, Mond and Pellikaan describe the second Fitting ideal as the ideal quotient of the square of the first Fitting ideal by the zeroth Fitting ideal. In this thesis, we generalize this result to iteratively construct higher Fitting ideals in the case of corank 1 map-germs. We also present a proof of the same result for matrices with generic entries, suggesting the possibility of further generalization to other cases such as corank ≥ 2, and show that a related result of Piene, established over the source of f, also holds in the target.

Recommended Citation

Smith, Jacob L., "FITTING IDEALS WITHOUT A PRESENTATION" (2024). Open Access Dissertations. Paper 1725.
https://digitalcommons.uri.edu/oa_diss/1725