The Aluffi Algebra and Linearity Condition

The Aluffi algebra is an algebraic version of characteristic cycles in intersection theory which is an intermediate graded algebra between the symmetric algebra (naive blowup) and the Rees algebra (blowup). Let  R be a commutative Noetherian ring and J ⊂I  ideals of R. We say that J ⊂I  satisfy linearity condition if the Aluffi algebra of I/J is isomorphic with the symmetric algebra. In this paper, we present the necessary and sufficient conditions for satisfying linearity condition. We give classes of ideals J ⊂I   which satisfy linearity condition. We prove that if J is a complete intersection ideal contained in I, then    J ⊂I  satisfy linearity condition if and only if the ideal I is of linear type.