On the Gorenstein Property of the Fiber Cone to Filtrations

TL;DR

This paper investigates when the fiber cone of a Hilbert filtration in a Noetherian local ring is Gorenstein, providing necessary and sufficient conditions based on algebraic invariants and describing the canonical module.

Contribution

It establishes criteria for the Gorenstein property of the fiber cone in terms of lengths and generators, extending known results from the $I$-adic case to Hilbert filtrations.

Findings

Gorenstein criteria for the fiber cone under Cohen-Macaulay assumptions

Explicit description of the canonical module of the fiber cone

Upper bound on the multiplicity of the canonical module

Abstract

Let $(A, \mathfrak{m})$ be a Noetherian local ring and $\mathfrak{F}=(I_{n})_{n\geq 0}$ a filtration. In this paper, we study the Gorenstein properties of the fiber cone $F(\mathfrak{F})$, where $\mathfrak{F}$ is a Hilbert filtration. Suppose that $F(\mathfrak{F})$ and $G(\mathfrak{F})$ are Cohen-Macaulay. If in addition, the associated graded ring $G(\mathfrak{F})$ is Gorenstein; similarly to the $I$-adic case, we obtain a necessary and sufficient condition, in terms of lengths and minimal number of generators of ideals, for Gorensteiness of the fiber cone. Moreover, we find a description of the canonical module of $F(\mathfrak{F})$ and show that even in the Hilbert filtration case, the multiplicity of the canonical module of the fiber cone is upper bounded by multiplicity of the canonical modules of the associated graded ring.