Monomial Algebras

Description

Monomial Algebras, Second Edition presents algebraic, combinatorial, and computational methods for studying monomial algebras and their ideals, including Stanley–Reisner rings, monomial subrings, Ehrhart rings, and blowup algebras. It emphasizes square-free monomials and the corresponding graphs, clutters, or hypergraphs.

New to the Second Edition

• Four new chapters that focus on the algebraic properties of blowup algebras in combinatorial optimization problems of clutters and hypergraphs
• Two new chapters that explore the algebraic and combinatorial properties of the edge ideal of clutters and hypergraphs
• Full revisions of existing chapters to provide an up-to-date account of the subject

Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs. It directly links the algebraic properties of monomial algebras to combinatorial structures (such as simplicial complexes, posets, digraphs, graphs, and clutters) and linear optimization problems.

Table of Contents

1. Polyhedral Geometry and Linear Optimization
   1. Polyhedral sets and cones
   2. Relative volumes of lattice polytopes
   3. Hilbert bases and TDI systems
   4. Rees cones and clutters
   5. The integral closure of a semigroup
   6. Unimodularity of matrices and normality
   7. Normaliz, a computer program
   8. Cut-incidence matrices and integrality
   9. Elementary vectors and matroids
      
2. Commutative Algebra
   1. Module theory
   2. Graded modules and Hilbert polynomials
   3. Cohen–Macaulay modules
   4. Normal rings
   5. Valuation rings
   6. Krull rings
   7. Koszul homology
   8. A vanishing theorem of Grothendieck
      
3. Affine and Graded Algebras
   1. Cohen–Macaulay graded algebras
   2. Hilbert Nullstellensatz
   3. Gröbner bases
   4. Projective closure
   5. Minimal resolutions
      
4. Rees Algebras and Normality
   1. Symmetric algebras
   2. Rees algebras and syzygetic ideals
   3. Complete and normal ideals
   4. Multiplicities and a criterion of Herzog
   5. Jacobian criterion
      
5. Hilbert Series
   1. Hilbert–Serre Theorem
      
   2. a-invariants and h-vectors
   3. Extremal algebras
   4. Initial degrees of Gorenstein ideals
   5. Koszul homology and Hilbert functions
   6. Hilbert functions of some graded ideals
      
6. Stanley–Reisner Rings and Edge Ideals of Clutters
   1. Primary decomposition
   2. Simplicial complexes and homology
   3. Stanley–Reisner rings
   4. Regularity and projective dimension
   5. Unmixed and shellable clutters
   6. Admissible clutters
   7. Hilbert series of face rings
   8. Simplicial spheres
   9. The upper bound conjectures
      
7. Edge Ideals of Graphs
   1. Graph theory
   2. Edge ideals and B-graphs
   3. Cohen–Macaulay and chordal graphs
   4. Shellable and sequentially C–M graphs
   5. Regularity, depth, arithmetic degree
   6. Betti numbers of edge ideals
   7. Associated primes of powers of ideals
      
8. Toric Ideals and Affine Varieties
   1. Binomial ideals and their radicals
   2. Lattice ideals
   3. Monomial subrings and toric ideals
   4. Toric varieties
   5. Affine Hilbert functions
   6. Vanishing ideals over finite fields
   7. Semigroup rings of numerical semigroups
   8. Toric ideals of monomial curves
      
9. Monomial Subrings
   1. Integral closure of monomial subrings
   2. Homogeneous monomial subrings
   3. Ehrhart rings
   4. The degree of lattice and toric ideals
   5. Laplacian matrices and ideals
   6. Gröbner bases and normal subrings
   7. Toric ideals generated by circuits
   8. Divisor class groups of semigroup rings
      
10. Monomial Subrings of Graphs
    1. Edge subrings and ring graphs
    2. Incidence matrices and circuits
    3. The integral closure of an edge subring
    4. Ehrhart rings of edge polytopes
    5. Integral closure of Rees algebras
    6. Edge subrings of complete graphs
    7. Edge cones of graphs
    8. Monomial birational extensions
       
11. Edge Subrings and Combinatorial Optimization
    1. The canonical module of an edge subring
    2. Integrality of the shift polyhedron
    3. Generators for the canonical module
    4. Computing the a-invariant
    5. Algebraic invariants of edge subrings
       
12. Normality of Rees Algebras of Monomial Ideals
    1. Integral closure of monomial ideals
    2. Normality criteria
    3. Rees cones and polymatroidal ideals
    4. Veronese subrings and the a-invariant
    5. Normalizations of Rees algebras
    6. Rees algebras of Veronese ideals
    7. Divisor class group of a Rees algebra
    8. Stochastic matrices and Cremona maps
       
13. Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters
    1. Vertex covers of clutters
    2. Symbolic Rees algebras of edge ideals
    3. Blowup algebras in perfect graphs
    4. Algebras of vertex covers of graphs
    5. Edge subrings in perfect matchings
    6. Rees cones and perfect graphs
    7. Perfect graphs and algebras of covers
       
14. Combinatorial Optimization and Blowup Algebras
    1. Blowup algebras of edge ideals
    2. Rees algebras and polyhedral geometry
    3. Packing problems and blowup algebras
    4. Uniform ideal clutters
    5. Clique clutters of comparability graphs
    6. Duality and integer rounding problems
    7. Canonical modules and integer rounding
    8. Clique clutters of Meyniel graphs
15. Appendix: Graph Diagrams
16. Bibliography
17. Notation Index
18. Index

Reviews

“… an introduction to algebraic, combinatorial, and computational aspects of monomial ideals. In the second edition, a full revision of all the chapters has been made.”

—Zentralblatt MATH 1325

Author Description

Dr. Rafael H. Villarreal is a professor in the Department of Mathematics at the Centro de Investigación y de Estudios Avanzados del I.P.N. (Cinvestav-IPN). His research focuses on commutative algebra, algebraic geometry, combinatorics, and computational algebra.