Exact pairs of homogeneous zero divisors

TL;DR

This paper investigates the constraints on the Hilbert function of standard graded Artinian algebras imposed by the presence of exact pairs of homogeneous zero divisors, showing that compressed level algebras lack such divisors.

Contribution

It establishes new restrictions on Hilbert functions for algebras with exact zero divisor pairs and proves that compressed level algebras cannot have homogeneous zero divisors.

Findings

Constraints on Hilbert functions from zero divisor pairs

Compressed level algebras have no homogeneous zero divisors

Identification of algebraic conditions related to zero divisors

Abstract

Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that if S is a compressed level algebra, then S does not contain any homogeneous zero divisors.