Rings for which general linear forms are exact zero divisors

TL;DR

This paper studies certain graded algebras over fields of characteristic zero where general linear forms act as exact zero divisors, proposing a conjecture about their Hilbert functions and proving it in specific cases.

Contribution

It introduces a conjecture on the Hilbert function of these algebras and proves it for monomial quotient rings and certain quadratic ideals.

Findings

Conjecture formulated for Hilbert functions of these rings.

Proved conjecture for monomial quotient rings.

Proved conjecture for quadratic ideals with mostly monomial generators.

Abstract

We investigate the standard graded $k$-algebras over a field $k$ of characteristic zero for which general linear forms are exact zero divisors. We formulate a conjecture regarding the Hilbert function of such rings. We prove our conjecture in the case when the ring is a quotient of a polynomial ring by a monomial idea, and also in the case when the ideal is generated in degree 2 and all but one of the generators are monomials.