Torsion-free Aluffi algebras

A pair of ideals J subset of I subset of R has been called Aluffi torsion-free if the Aluffi algebra of I/J is isomorphic to 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) subset of I such that J(1) - J(2) epsilon 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. (C) 2018 Elsevier Inc. All rights reserved.