704 Pages 106 B/W Illustrations
    by Chapman & Hall

    704 Pages 106 B/W Illustrations
    by Chapman & Hall

    704 Pages 106 B/W Illustrations
    by Chapman & Hall

    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.

    Polyhedral Geometry and Linear Optimization
    Polyhedral sets and cones
    Relative volumes of lattice polytopes
    Hilbert bases and TDI systems
    Rees cones and clutters
    The integral closure of a semigroup
    Unimodularity of matrices and normality
    Normaliz, a computer program
    Cut-incidence matrices and integrality
    Elementary vectors and matroids

    Commutative Algebra
    Module theory
    Graded modules and Hilbert polynomials
    Cohen–Macaulay modules
    Normal rings
    Valuation rings
    Krull rings
    Koszul homology
    A vanishing theorem of Grothendieck

    Affine and Graded Algebras
    Cohen–Macaulay graded algebras
    Hilbert Nullstellensatz
    Gröbner bases
    Projective closure
    Minimal resolutions

    Rees Algebras and Normality
    Symmetric algebras
    Rees algebras and syzygetic ideals
    Complete and normal ideals
    Multiplicities and a criterion of Herzog
    Jacobian criterion

    Hilbert Series
    Hilbert–Serre Theorem
    a-invariants and h-vectors
    Extremal algebras
    Initial degrees of Gorenstein ideals
    Koszul homology and Hilbert functions
    Hilbert functions of some graded ideals

    Stanley–Reisner Rings and Edge Ideals of Clutters
    Primary decomposition
    Simplicial complexes and homology
    Stanley–Reisner rings
    Regularity and projective dimension
    Unmixed and shellable clutters
    Admissible clutters
    Hilbert series of face rings
    Simplicial spheres
    The upper bound conjectures

    Edge Ideals of Graphs
    Graph theory
    Edge ideals and B-graphs
    Cohen–Macaulay and chordal graphs
    Shellable and sequentially C–M graphs
    Regularity, depth, arithmetic degree
    Betti numbers of edge ideals
    Associated primes of powers of ideals

    Toric Ideals and Affine Varieties
    Binomial ideals and their radicals
    Lattice ideals
    Monomial subrings and toric ideals
    Toric varieties
    Affine Hilbert functions
    Vanishing ideals over finite fields
    Semigroup rings of numerical semigroups
    Toric ideals of monomial curves

    Monomial Subrings
    Integral closure of monomial subrings
    Homogeneous monomial subrings
    Ehrhart rings
    The degree of lattice and toric ideals
    Laplacian matrices and ideals
    Gröbner bases and normal subrings
    Toric ideals generated by circuits
    Divisor class groups of semigroup rings

    Monomial Subrings of Graphs
    Edge subrings and ring graphs
    Incidence matrices and circuits
    The integral closure of an edge subring
    Ehrhart rings of edge polytopes
    Integral closure of Rees algebras
    Edge subrings of complete graphs
    Edge cones of graphs
    Monomial birational extensions

    Edge Subrings and Combinatorial Optimization
    The canonical module of an edge subring
    Integrality of the shift polyhedron
    Generators for the canonical module
    Computing the a-invariant
    Algebraic invariants of edge subrings

    Normality of Rees Algebras of Monomial Ideals
    Integral closure of monomial ideals
    Normality criteria
    Rees cones and polymatroidal ideals
    Veronese subrings and the a-invariant
    Normalizations of Rees algebras
    Rees algebras of Veronese ideals
    Divisor class group of a Rees algebra
    Stochastic matrices and Cremona maps

    Combinatorics of Symbolic Rees Algebras of Edge Ideals of Clutters
    Vertex covers of clutters
    Symbolic Rees algebras of edge ideals
    Blowup algebras in perfect graphs
    Algebras of vertex covers of graphs
    Edge subrings in perfect matchings
    Rees cones and perfect graphs
    Perfect graphs and algebras of covers

    Combinatorial Optimization and Blowup Algebras
    Blowup algebras of edge ideals
    Rees algebras and polyhedral geometry
    Packing problems and blowup algebras
    Uniform ideal clutters
    Clique clutters of comparability graphs
    Duality and integer rounding problems
    Canonical modules and integer rounding
    Clique clutters of Meyniel graphs

    Appendix: Graph Diagrams


    Notation Index



    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.

    "… 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