On decomposing monomial algebras with the Lefschetz properties

TL;DR

This paper introduces a new technique for decomposing monomial algebras to analyze their Lefschetz properties, demonstrating its effectiveness on various algebra classes including Gorenstein algebras from numerical semigroups.

Contribution

The paper presents a novel decomposition method for monomial algebras and applies it to establish Lefschetz properties for new classes of algebras, including Gorenstein codimension three cases.

Findings

Gorenstein codimension three algebras from numerical semigroups have the strong Lefschetz property

A decomposition technique effectively analyzes Lefschetz properties in monomial algebras

Gluing operations can construct monomial algebras with Lefschetz properties

Abstract

We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. We apply our technique to various classes of algebras, including monomial almost complete intersections and Gorenstein algebras. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property. We also study the reverse of the splitting operation -- a gluing operation -- which gives a way to construct monomial algebras with the Lefschetz properties.