# Torsion-free Aluffi Algebras

## TL;DR

This paper characterizes when the Aluffi algebra of a quotient of ideals is isomorphic to the Rees algebra, providing conditions based on syzygy modules and introducing strongly Aluffi torsion-free ideals.

## Contribution

It offers necessary and sufficient conditions for Aluffi torsion-freeness using syzygy modules and introduces the concept of strongly Aluffi torsion-free ideals.

## Key findings

- Criteria for Aluffi torsion-freeness in terms of syzygy modules.
- Equivalence conditions for pairs of ideals with similar form ideals.
- Introduction and initial results on strongly Aluffi torsion-free ideals.

## Abstract

A pair of ideals $J\subseteq I\subseteq R$ has been called Aluffi torsion-free if the Aluffi algebra of $I/J$ is isomorphic with the corresponding Rees algebra. We give necessary and sufficient conditions for the Aluffi torsion-free property in terms of the first syzygy module of the form ideal $J^*$ in the associated graded ring of $I$. For two pairs of ideals $J_1,J_2\subseteq I$ such that $J_1-J_2\in I^2$, we prove that if one pair is Aluffi torsion-free the other one is so if and only if the first syzygy modules of $J_1$ and $J_2$ have the same form ideals. We introduce the notion of strongly Aluffi torsion-free ideals and present some results on these ideals.