Theory of tangential idealizers and tangentially free ideals

TL;DR

This paper extends the theory of free divisors by introducing and analyzing tangential idealizers and tangentially free ideals, providing criteria for their reflexiveness and freeness in factorial domains.

Contribution

It generalizes the theory of logarithmic derivations by defining tangential idealizers and establishing conditions for their reflexiveness and freeness, expanding the concept of free divisors.

Findings

Established reflexiveness criteria for tangential idealizers in factorial domains

Derived necessary and sufficient conditions for their freeness

Introduced the class of tangentially free ideals

Abstract

We generalize the theory of logarithmic derivations through a self-contained study of modules here dubbed tangential idealizers. We establish reflexiveness criteria for such modules, provided the ring is a factorial domain. As a main consequence, necessary e sufficient conditions for their freeness are derived and the class of tangentially free ideals is introduced, thus extending (algebraically) the theory of free divisors proposed by K. Saito around 30 years ago.