The Gorenstein and Complete Intersection Properties of Associated Graded Rings

Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F (I) of I as reflected in their defining ideals as homomorphic images of polynomial rings over R/I and R/m respectively. In case all the higher conormal modules of I are free over R/I , we observe that: (i) G(I) is Cohen-Macaulay iff F (I) is Cohen-Macaulay, (ii) G(I) is Gorenstein iff both F (I) and R/I are Gorenstein, and (iii) G(I) is a relative complete intersection iff F (I) is a complete intersection. In case (R,m) is Gorenstein, we give a necessary and sufficient condition for G(I) to be Gorenstein in terms of residuation of powers of I with respect to a reduction J of I with μ(J) = dimR and the reduction number r of I with respect to J . We prove that G(I) is Gorenstein if and only if J : I = J + I for 0 ≤ i ≤ r − 1. If (R,m) is a Gorenstein local ring and I ⊆ m is an ideal having a reduction J with reduction number r such that μ(J) = ht(I) = g > 0, we prove that the extended Rees algebra R[It, t] is quasi-Gorenstein with a-invariant a if and only if J i : I = I for every i ∈ Z. If, in addition, dimR = 1, we show that G(I) is Gorenstein if and only if J i : I = I for 1 ≤ i ≤ r. Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary: 13A30; Secondary: 13C40, 13H10, 13H15.