Date of Award

Spring 2026

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Adela Vraciu

Abstract

In 1980, Richard Stanley proved that Artinian monomial complete intersections enjoy the strong Lefschetz property in characteristic zero. Since his landmark result, the strong and weak Lefschetz properties (SLP and WLP) have garnered attention from an increasing number of mathematicians from an ever-broadening swath of the mathematical community. Indeed, the Lefschetz properties have been scrutinized for their relation to algebraic invariants such as Betti numbers and the Hilbert function as well as their links from commutative algebra to other branches of mathematics.

However, the Lefschetz properties are also quite subtle; the aforementioned result due to Stanely is a rather rare find in the sense that such sweeping classifications of WLP (and/or SLP) are generally hard to come by. Many classes of ideals have been examined in hopes of classifying when the WLP holds for the algebras they define. This dissertation aims to advance our understanding of the WLP for two different kinds of almost complete intersection ideals.

Chapter 1 gathers the necessary background and provides an overview of the WLP before offering a handful of disparate stand-alone results. In Chapter 2, we bound the earliest degree for which multiplication by a general linear form can fail to be injective for almost complete intersections generated by a uniform power of general linear forms. We improve on the general bound when the defining ideal is generated by squares and cubes, as well as apply our results to Fröberg’s conjecture. Finally, Chapter 3 determines the generators of a particular colon ideal in two variables and applies them toward resolving some open cases of the WLP for monomial almost complete intersections in three variables.

Rights

© 2026, Matthew Davidson Booth

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