Monomial Ideals and Their Decompositions

Contents
Introduction
	Overview
	Audience
	Summary of Contents
	Notes for the Instructor/Independent Reader
	Possible Course Outlines
	Acknowledgments
Part I Monomial Ideals
1 Fundamental Properties of Monomial Ideals
	1.1 Monomial Ideals
	1.2 Integral Domains (optional)
	1.3 Generators of Monomial Ideals
	1.4 Noetherian Rings (optional)
	1.5 Exploration: Counting Monomials
	1.6 Exploration: Numbers of Generators
2 Operations on Monomial Ideals
	2.1 Intersections of Monomial Ideals
	2.2 Unique Factorization Domains (optional)
	2.3 Monomial Radicals
	2.4 Exploration: Reduced Rings
	2.5 Colons of Monomial Ideals
	2.6 Bracket Powers of Monomial Ideals
	2.7 Exploration: Saturation
	2.8 Exploration: Generalized Bracket Powers
	2.9 Exploration: Comparing Bracket Powers and Ordinary Powers
3 M-Irreducible Ideals and Decompositions
	3.1 M-Irreducible Monomial Ideals
	3.2 Irreducible Ideals (optional)
	3.3 M-Irreducible Decompositions
	3.4 Irreducible Decompositions (optional)
	3.5 Exploration: Decompositions in Two Variables, part I
Part II Monomial Ideals and Other Areas
4 Connections with Combinatorics
	4.1 Square-Free Monomial Ideals
	4.2 Graphs and Edge Ideals
	4.3 Decompositions of Edge Ideals
	4.4 Simplicial Complexes and Stanley-Reisner Ideals
	4.5 Decompositions of Stanley-Reisner Ideals
	4.6 Facet Ideals and Their Decompositions
	4.7 Exploration: Alexander Duality
5 Connections with Other Areas
	5.1 Krull Dimension
	5.2 Vertex Covers and PMU Placement
	5.3 Cohen-Macaulayness and the Upper Bound Theorem
	5.4 Hilbert Functions and Initial Ideals
	5.5 Resolutions of Monomial Ideals
Part III Decomposing Monomial Ideals
6 Parametric Decompositions of Monomial Ideals
	6.1 Parameter Ideals and Parametric Decompositions
	6.2 Corner Elements
	6.3 Finding Corner Elements in Two Variables
	6.4 Finding Corner Elements in General
	6.5 Exploration: Decompositions in Two Variables, part II
	6.6 Exploration: Decompositions of Some Powers of Ideals
	6.7 Exploration: Macaulay Inverse Systems
7 Computing M-Irreducible Decompositions
	7.1 M-Irreducible Decompositions of Monomial Radicals
	7.2 M-Irreducible Decompositions of Bracket Powers
	7.3 M-Irreducible Decompositions of Sums
	7.4 M-Irreducible Decompositions of Colon Ideals
	7.5 Computing General M-Irreducible Decompositions
	7.6 Exploration: Edge, Stanley-Reisner, and Facet Ideals Revisited
	7.7 Exploration: Decompositions of Saturations
	7.8 Exploration: Decompositions of Generalized Bracket Powers
	7.9 Exploration: Decompositions of Products of Monomial Ideals
Part IV Commutative Algebra and Macaulay2
Appendix A: Foundational Concepts
	A.1 Rings
	A.2 Polynomial Rings
	A.3 Ideals and Generators
	A.4 Sums of Ideals
	A.5 Products and Powers of Ideals
	A.6 Colon Ideals
	A.7 Radicals of Ideals
	A.8 Quotient Rings
	A.9 Partial Orders and Monomial Orders
	A.10 Exploration: Algebraic Geometry
Appendix B: Introduction to Macaulay2
	B.1 Rings
	B.2 Polynomial Rings
	B.3 Ideals and Generators
	B.4 Sums of Ideals
	B.5 Products and Powers of Ideals
	B.6 Colon Ideals
	B.7 Radicals of Ideals
	B.8 Quotient Rings
	B.9 Monomial Orders
Further Reading
References
Index of Macaulay2 Commands, by Command
Index of Macaulay2 Commands, by Description
Index of Names
Index of Symbols
Index of Terminology