Multigraded Betti Numbers of Squarefree Monomial Ideals from Maximal, Nonattacking Rook Arrangements
Faculty Mentor
Dr. Tricia Brown
Location
Russell Union Ballroom
Type of Research
Proposed
Session Format
Poster Presentation
College
College of Science & Mathematics
Department
Mathematics
Abstract
This project investigates algebraic structures that arise from combinatorial constructions using tools from commutative algebra, combinatorics, and topology. Many natural combinatorial objects give rise to squarefree, pure monomial ideals. One can construct minimal free resolutions of these ideals and study their graded and multigraded Betti numbers, which measure the complexity of the resolution. The connection between monomial ideals and Betti numbers is provided by Hochster’s formula, which expresses the Betti numbers of a squarefree monomial ideal in terms of the reduced homology of associated simplicial complexes.
The primary goal of this work is to develop combinatorial interpretations of these Betti numbers by relating them to counts of structures inherent in the original combinatorial objects. Understanding these relationships provides insight into the invariants of the associated ideals and their resolutions. Ultimately, this project aims to contribute toward a systematic and efficient method for constructing minimal free resolutions of monomial ideals that arise from combinatorial objects.
As a central case study, we examine monomial ideals generated by maximal, nonattacking rook placements on n×n chessboards. These configurations naturally produce squarefree monomial ideals whose algebraic invariants reflect properties of the underlying combinatorial arrangements. To analyze these ideals, we employ a theorem that relates a simplicial complex with its associated subcomplexes through the operations of deletion and link, allowing us to study properties of higher-dimensional complexes through the lens of lower-dimensional ones.
Through this framework, we investigate patterns in the Betti tables of the corresponding ideals. In particular, we conjecture that the entries in the first column of the Betti table correspond precisely to products of two monomials.
Program Description
n/a
DOI
10.20429/GS4.2026.027
Start Date
4-23-2026 2:00 PM
End Date
4-23-2026 4:00 PM
Recommended Citation
Flaherty, Gwendolyn, "Multigraded Betti Numbers of Squarefree Monomial Ideals from Maximal, Nonattacking Rook Arrangements" (2026). GS4 Student Scholars Symposium. 224.
https://digitalcommons.georgiasouthern.edu/research_symposium/2026/2026/224
Multigraded Betti Numbers of Squarefree Monomial Ideals from Maximal, Nonattacking Rook Arrangements
Russell Union Ballroom
This project investigates algebraic structures that arise from combinatorial constructions using tools from commutative algebra, combinatorics, and topology. Many natural combinatorial objects give rise to squarefree, pure monomial ideals. One can construct minimal free resolutions of these ideals and study their graded and multigraded Betti numbers, which measure the complexity of the resolution. The connection between monomial ideals and Betti numbers is provided by Hochster’s formula, which expresses the Betti numbers of a squarefree monomial ideal in terms of the reduced homology of associated simplicial complexes.
The primary goal of this work is to develop combinatorial interpretations of these Betti numbers by relating them to counts of structures inherent in the original combinatorial objects. Understanding these relationships provides insight into the invariants of the associated ideals and their resolutions. Ultimately, this project aims to contribute toward a systematic and efficient method for constructing minimal free resolutions of monomial ideals that arise from combinatorial objects.
As a central case study, we examine monomial ideals generated by maximal, nonattacking rook placements on n×n chessboards. These configurations naturally produce squarefree monomial ideals whose algebraic invariants reflect properties of the underlying combinatorial arrangements. To analyze these ideals, we employ a theorem that relates a simplicial complex with its associated subcomplexes through the operations of deletion and link, allowing us to study properties of higher-dimensional complexes through the lens of lower-dimensional ones.
Through this framework, we investigate patterns in the Betti tables of the corresponding ideals. In particular, we conjecture that the entries in the first column of the Betti table correspond precisely to products of two monomials.
