This book provides an introduction to combinatorial commutative algebra with particular emphasis on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determined rings. The eighteen chapters cover topics ranging from homological invariants of monomial ideals and their polyhedral resolutions, to hands-on tools for studying algebraic varieties with group actions, such as toric varieties, flag varieties, quiver loci, and Hilbert schemes. Contains over 100 figures and 250 exercises.

Klappentext

Recent developments are covered

Contains over 100 figures and 250 exercises

Includes complete proofs

Inhalt

Monomial Ideals.- Squarefree monomial ideals.- Borel-fixed monomial ideals.- Three-dimensional staircases.- Cellular resolutions.- Alexander duality.- Generic monomial ideals.- Toric Algebra.- Semigroup rings.- Multigraded polynomial rings.- Syzygies of lattice ideals.- Toric varieties.- Irreducible and injective resolutions.- Ehrhart polynomials.- Local cohomology.- Determinants.- Plücker coordinates.- Matrix Schubert varieties.- Antidiagonal initial ideals.- Minors in matrix products.- Hilbert schemes of points.