Theses and Dissertations

Date of Award

5-1-2025

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics

First Advisor

Luigi Ferraro

Second Advisor

Debanjana Kundu

Third Advisor

Timothy Huber

Abstract

This thesis investigates the construction of a DG Γ-algebra structure on the Generalized Taylor Resolution (GTR) associated with monomial ideals. The classical Taylor resolution is known for providing a free but generally non-minimal resolution, leading to computational challenges and inefficiencies in algebraic analysis. In contrast, the GTR preserves essential algebraic structures while optimizing the resolution process, offering a more efficient and comprehensive framework for studying monomial ideals.

We introduce a novel DG Γ-structure that incorporates divided powers into the GTR, enhancing its multiplicative and homological properties. This structure preserves strict graded commutativity and is fully compatible with the differential graded framework, thus facilitating a more nuanced exploration of the algebraic and homological properties of monomial ideals. Our approach builds on the foundational contributions of Avramov, Herzog, and VandeBogert, whose pioneering work laid the groundwork for understanding DG Γ-algebras. By extending these foundational theories, this research explores more complex algebraic scenarios, contributing to a comprehensive and rigorous approach to monomial resolutions.

The methodology involves constructing the DG Γ-structure on the GTR ensuring its compatibility with divided power operations and graded commutativity requirements. Through careful construction and proof, we establish the uniqueness of this DG Γ-algebra structure on the GTR, thus preserving the homological integrity of the resolution. The unique structure developed in this thesis facilitates the construction of multigraded free resolutions, which are particularly effective for resolving sums of monomial ideals, a scenario commonly encountered in commutative algebra and algebraic geometry. Additionally, we provide explicit formulations for the differential and product maps, guaranteeing the coherence and consistency of the divided power operations within this algebraic context.

To provide a comprehensive understanding, this work explores how this structure influences the Taylor complex’s multiplicative properties, thereby enriching the algebraic framework used for studying monomial resolutions. By leveraging the generalized Taylor complex, the study presents a cohesive and systematic approach to understanding the structural properties of the GTR.

The findings presented in this thesis enhance the understanding of the properties of the GTR and provide a new perspective on its structural characteristics. By advancing the theoretical framework for DG Γ-algebras, this work contributes to the algebraic toolkit for analyzing complex monomial resolutions and offers a more detailed view of the generalized Taylor complexes. These contributions not only provide a deeper insight into the algebraic and homological nature of monomial ideals but also pave the way for further research and potential applications in computational algebra, homological algebra, and related fields of mathematical study.

Comments

Copyright 2025 Raul F Alvarez. https://proquest.com/docview/3240606200

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