On Gorensteinness of associated graded rings of filtrations

2024/04/22 by Meghana Bhat, Bhat, Meghana, Saipriya Dubey +11 · 1 citation
Mathematics · #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2404.14189

Abstract

Let (A, \mathfrakm) be a Gorenstein local ring, and F =\F_n \_{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,…, x_m] ] is a formal power series ring over an algebraically closed field K, and f=x₀^a-g(x₁,…,x_m), where g is a polynomial with g ∈ (x₁,…,x_m)^b ∖ (x₁,…,x_m)^b+1, and a, b, m are integers. We show that the normal tangent cone G(\mathfrakm) is Cohen-Macaulay if A is normal and a ≤ b. Moreover, we give a criterion of the Gorensteinness of G(\mathfrakm).

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