On Gorensteiness of associated graded rings of filtrations

Let (A, m) be a Gorenstein local ring, and F =ₙ ∈ Z 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 Ḡ(m) is Cohen-Macaulay if A is normal and a ≤ b. Moreover, we give a criterion of the Gorensteinness of Ḡ(m).