2nd Edition

Monomial Algebras





ISBN 9781138894181
Published March 12, 2018 by Chapman and Hall/CRC
704 Pages 106 B/W Illustrations

USD $86.95

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Book 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

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

Bibliography

Notation Index

Index

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Author(s)

Biography

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.

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