Specialization and Integral Closure

TL;DR

This paper investigates how integral closure properties of ideals and modules behave under specialization, enabling inductive proofs and simplifying complex problems in commutative algebra.

Contribution

It establishes the compatibility of integral closedness with generic specialization for ideals of height at least two and extends results to modules via Bourbaki ideals.

Findings

Integral closedness of ideals of height ≥ 2 is compatible with generic specialization.

An element's integrality over a module can be checked modulo a generic element.

Results facilitate inductive proofs and reduce module problems to ideal problems.

Abstract

We prove that the integral closedness of any ideal of height at least two is compatible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional assumptions, we show that an element is integral over a module if it is integral modulo a generic element of the module. This turns questions about integral closures of modules into problems about integral closures of ideals, by means of a construction known as Bourbaki ideal.