One-dimensional Gorenstein local rings with decreasing Hilbert function

TL;DR

This paper constructs infinitely many one-dimensional Gorenstein local rings with decreasing Hilbert functions at specified levels, demonstrating that such decreases are unbounded and providing new insights into their structure.

Contribution

It provides explicit constructions of Gorenstein local rings with decreasing Hilbert functions at almost all levels, solving a longstanding problem and showing no bounds to the decrease.

Findings

Constructed infinitely many Gorenstein local rings with decreasing Hilbert functions.

Proved there are no bounds to the levels at which the Hilbert function decreases.

Included many explicit examples illustrating the phenomena.

Abstract

In this paper we solve a problem posed by M.E. Rossi: {\it Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? } More precisely, for any integer $h>1$, $h \notin\{14+22k, \, 35+46k \ | \ k\in\mathbb{N} \}$, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level $h$; moreover we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included.