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:Ir-i=J+Ii+1 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-1] is quasi-Gorenstein with a-invariant a if and only if Ji:Ir=Ii+a-r+g-1 for every i∈Z. If, in addition, dimR=1, we show that G(I) is Gorenstein if and only if Ji:Ir=Ii for 1⩽i⩽r.