Aluffi torsion-free ideals

TL;DR

This paper investigates Aluffi torsion-free ideals, a class of algebras between symmetric and Rees algebras, focusing on ideals generated by minors of matrices and graph edge ideals, providing conditions for torsion-freeness.

Contribution

It establishes conditions for Aluffi torsion-free property in specific classes of ideals, such as minors of matrices and graph edge ideals, expanding understanding of these algebraic structures.

Findings

Conditions equivalent to Aluffi torsion-free property for certain ideals.

Characterization of torsion-freeness in ideals generated by minors.

Examples illustrating the Aluffi torsion-free property.

Abstract

A special class of algebras which are intermediate between the symmetric and the Rees algebras of an ideal was introduced by P. Aluffi in 2004 to define characteristic cycle of a hypersurface parallel to conormal cycle in intersection theory. These algebras are recently investigated by A. Nasrollah Nejad and A. Simis who named them Aluffi algebras. For a pair of ideals $J\subseteq I$ of a commutative ring $R$, the Aluffi algebra of $I/J$ is called Aluffi torsion-free if it is isomorphic to the Rees algebra of $I/J$. In this paper, ideals generated by 2-minors of a $2\times n$ matrix of linear forms and also edge ideals of graphs are considered and some conditions are presented which are equivalent to Aluffi torsion-free property of them. Also many other examples and further questions are presented.