Monomial algebras and polyhedral geometry

Abstract

This chapter discusses the monomial algebras and its connections to combinatorics, graph theory, and polyhedral geometry. Some important notions from commutative algebra that have played a role in the development of the theory, such as Cohen-Macaulay ring, normal ring, Gorenstein ring, integral closure, Hilbert series, and local cohomology are introduced. The upper bound theorem for the number of faces of a simplicial sphere, a description of the integral closure of an edge subring, a generalized marriage theorem for a certain family of graphs, and a study of systems of binomials in the ideal of an affine toric variety are provided as applications. It illustrates the interplay between several areas of mathematics and the power of combinatorial commutative algebra techniques. There is a connection between monomial rings and monomial subrings due to the fact that the initial ideal of a toric ideal is a monomial ideal. This allows computing several invariants of projective varieties using algebraic systems such as CoCoA and Macaulay2. An important tool to study monomial subrings is Normaliz, which is effective in practice and can be used to find normalizations, Hilbert series, Ehrhart rings, and volumes of lattice polytopes.