On Gorensteinness of associated graded rings of filtrations
Meghana Bhat, Saipriya Dubey, Shreedevi K. Masuti, Tomohiro Okuma, Jugal K. Verma, Kei-ichi Watanabe, Ken-ichi Yoshida
TL;DR
The paper develops criteria for when the associated graded ring $G(\mathcal{F})$ of a Hilbert filtration is Gorenstein, articulating conditions in terms of Hilbert coefficients and reduction numbers and extending results of Okuma–Watanabe–Yoshida to higher dimensions and normal filtrations. It analyzes Zariski-type hypersurface rings to determine when the normal tangent cone $\overline{G}(\mathfrak m)$ is Cohen–Macaulay and when it is Gorenstein, providing explicit arithmetic criteria such as $b \equiv 0$ or $d \pmod{a}$ (with $d=\gcd(a,b)$). A key methodological contribution is the introduction of maximal relative reduction numbers to compute Hilbert series in challenging cases and to derive Gorenstein criteria via $h$-vector symmetry. The results illuminate the structure of tangent cones in singularities, give practical tests for Gorensteinness in both standard and normal filtrations, and unify several strands of prior work on Brieskorn/Zariski hypersurface rings.
Abstract
Let $(A, \mathfrak{m})$ be a Gorenstein local ring, and $\mathcal{F} =\{F_n \}_{n\in \mathbb{Z}}$ a Hilbert filtration. In this paper, we give a criterion for Gorensteinness of the associated graded ring of $\mathcal{F}$ in terms of the Hilbert coefficients of $\mathcal{F}$ 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 $S=K[\![x_0,x_1,\ldots, x_m]\!]$ is a formal power series ring over an algebraically closed field $K$, and $f=x_0^a-g(x_1,\ldots,x_m)$, where $g$ is a polynomial with $g \in (x_1,\ldots,x_m)^b \setminus (x_1,\ldots,x_m)^{b+1}$, and $a, \, b, \, m$ are integers. We show that the normal tangent cone $\overline{G}(\mathfrak{m})$ is Cohen-Macaulay if $A$ is normal and $a \le b$. Moreover, we give a criterion of the Gorensteinness of $\overline{G}(\mathfrak{m})$.