This book treats the interaction between discrete convex geometry, commutative ring theory, algebraic K-theory, and algebraic geometry. The basic mathematical objects are lattice polytopes, rational cones, affine monoids, the algebras derived from them, and toric varieties. The book discusses several properties and invariants of these objects, such as efficient generation, unimodular triangulations and covers, basic theory of monoid rings, isomorphism problems and automorphism groups, homological properties and enumerative combinatorics. The last part is an extensive treatment of the K-theory of monoid rings, with extensions to toric varieties and their intersection theory.

This monograph has been written with a view towards graduate students and researchers who want to study the cross-connections of algebra and discrete convex geometry. While the text has been written from an algebraist's view point, also specialists in lattice polytopes and related objects will find an up-to-date discussion of affine monoids and their combinatorial structure. Though the authors do not explicitly formulate algorithms, the book takes a constructive approach wherever possible.

Winfried Bruns is Professor of Mathematics at Universität Osnabrück.

Joseph Gubeladze is Professor of Mathematics at San Francisco State University.

Zusammenfassung

This book examines interactions of polyhedral discrete geometry and algebra. What makes this book different from others is the presentation of several central results in all three areas of the exposition - discrete geometry, commutative algebra, and K-theory. The only prerequisite for the reader is a background in algebra, and the basics of polyhedral geometry have been included in Chapter 1.

The text will be of interest to graduate students and mathematicians.  Included are numerous exercises, historical background, and notes throughout the chapters.

Inhalt
I Cones, monoids, and triangulations.- Polytopes, cones, and complexes.- Affine monoids and their Hilbert bases.- Multiples of lattice polytopes.- II Affine monoid algebras.- Monoid algebras.- Isomorphisms and automorphisms.- Homological properties and Hilbert functions.- Gr#x00F6;bner bases, triangulations, and Koszul algebras.- III K-theory.- Projective modules over monoid rings.- Bass#x2013;Whitehead groups of monoid rings.- Varieties.