Conormal modules via Primitive ideals

TL;DR

This paper investigates the structure of conormal modules and the second symbolic power of ideals in polynomial rings, providing new characterizations and applications in algebraic geometry.

Contribution

It introduces novel descriptions of the torsion part of conormal modules and characterizes when the torsion-free part is free, advancing understanding of ideal structures.

Findings

Expressed torsion part and torsion-free module via primitive ideals

Provided criteria for the torsion-free module to be free

Applied results to specific algebraic problems

Abstract

The main object of this note is to study the conormal module $M$ and the computation of the second symbolic power $\bar I^{(2)}$ of an ideal $\bar I$ in the residue ring $R/H$ of a polynomial ring $R$ over a field of characteristic zero. The torsion part $T(M)$ of $M$ and the torsion free module $M/T(M)$ are expressed by the primitive ideal of $I$ relative to $H$. Two characterizations for $M/T(M)$ to be free are proved. Some immediate applications are worked out.