On Gorensteiness of associated graded rings of filtrations

Let (A, m) be a Gorenstein local ring, and F =\Fₙ \₍ ₙ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of F in terms of the Hilbert coefficients of F in some cases. As a consequence we recover and extend a result proved by Okuma, Watanabe and Yoshida. Further, we present ring-theoretic properties of the normal tangent cone of the maximal ideal of A=S/ (f) where S=K\![x₀, x₁, , xₘ\!] is a formal power series ring over an algebraically closed field K, and f=x₀ᵃ-g (x₁, , xₘ), where g is a polynomial with g (x₁, , xₘ) ᵇ (x₁, , xₘ) ^b+1, and a, \, b, \, m are integers. We show that the normal tangent cone G (m) is Cohen-Macaulay if A is normal and a b. Moreover, we give a criterion of the Gorensteinness of G (m).