Gorenstein graded rings associated to ideals

Let A be a Noetherian local ring with the maximal ideal m and dim A = d. Let I(6= A) be an ideal in A and s = htA I. This article studies the question of when the associated graded ring G(I) := ⊕ i≥0 I /I is Gorenstein. To state our result, we set up some notation. Let ` be an integer such that ` ≤ d and let J be a reduction of I generated by elements a1, a2, . . . , a`. We denote by rJ(I) the reduction number of I with respect to J . The analytic spread of I is λ(I) := dim A/m ⊗A G(I). Then s ≤ λ(I) ≤ `. We always assume that the generating set {a1, a2, . . . , a`} of J is a basic generating set for J in the sense of Aberbach, Huneke, and Trung [AHT], which means that JiAq is a reduction of IAq for all q ∈ V(I) with i = htA q < `. Here let V(I) be a set of prime ideals in A containing I and Ji := (a1, a2, . . . , ai) for 0 ≤ i ≤ `. By [AHT], 7.2, there always exists a basic generating set for J if the field A/m is infinite. Let