This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals.

Providing a useful and quick introduction to areas of research spanning these fields, Monomial Ideals is split into three parts. Part I offers a quick introduction to the modern theory of Gröbner bases as well as the detailed study of generic initial ideals. Part II supplies Hilbert functions and resolutions and some of the combinatorics related to monomial ideals including the KruskalKatona theorem and algebraic aspects of Alexander duality. Part III discusses combinatorial applications of monomial ideals, providing a valuable overview of some of the central trends in algebraic combinatorics. Main subjects include edge ideals of finite graphs, powers of ideals, algebraic shifting theory and an introduction to discrete polymatroids. Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text.

Self-contained and concise, this book will appeal to a wide range of readers, including PhD students on advanced courses, experienced researchers, and combinatorialists and non-specialists with a basic knowledge of commutative algebra.

Since their first meeting in 1985, Juergen Herzog (Universität Duisburg-Essen, Germany) and Takayuki Hibi (Osaka University, Japan), have worked together on a number of research projects, of which recent results are presented in this monograph.

Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text Provides a quick and useful introduction to research spanning the fields of combinatorial and computational commutative algebra, with a special focus on monomial ideals Only a basic knowledge of commutative algebra is required, making this accessible to specialists and non-specialists alike

Autorentext

Jürgen Herzog was a professor of mathematics at the University of Duisburg-Essen, Germany. He received his doctorate at Louisiana State University in 1969 and completed his habilitation at the University of Regensburg in 1974. Since 1975, he was a professor at the University of Essen, later Duisburg-Essen, where he retired in 2009. His main research area was in the field of commutative algebra. During the course of his scientific career, he published more than 250 articles and made a significant impact in the development of commutative algebra. Jürgen Herzog sadly passed away on 23 April 2024 before this book could be published. He will be greatly missed in the mathematical community. Somayeh Moradi is associate professor of mathematics at Ilam University, Iran. She received her doctorate from Amirkabir University of Technology - Tehran Polytechnic in 2009. She was appointed assistant professor at Ilam University in 2010 and later in 2017 as associate professor. From march 2023 to March 2025, she was the recipient of the Alexander von Humboldt Research Fellowship at the University of Duisburg-Essen. She is also the recipient of the Maryam Mirzakhani Award 2025. Her main research interests lie in commutative algebra and combinatorics. Masoomeh Rahimbeigi is a researcher at the University of Duisburg-Essen, Germany. She received her doctorate from the University of Kurdistan, Iran in 2019. Her main research area is in commutative algebra.

Inhalt
Part I Gröbner bases: Monomial Ideals.- A short introduction to Gröbner bases.- Monomial orders and weights.- Generic initial ideals.- The exterior algebra.- Part II: Hilbert functions and resolutions.- Hilbert functions and the theorems of Macaulay and Kruskal-Katona.- Resolutions of monomial ideals and the Eliahou-Kervaire formula.- Alexander duality and resolutions.- Part III Combinatorics: Alexander duality and finite graphs.- Powers of monomial ideals.- Shifting theory.- Discrete Polymatroids.- Some homological algebra.- Geometry