This book examines interactions of polyhedral discrete geometry and algebra. What makes this book unique is the presentation of several central results in all three areas of the exposition - from discrete geometry, to commutative algebra, and K-theory.

Examines interactions of polyhedral discrete geometry and algebra Presents several central results in all three areas of the exposition-from discrete geometry, commutative algebra, and K-theory Has constructive (i.e. algorithmic) nature at many places throughout the text Despite the large amount of information from various fields, the polytopal perspective is kept as the major organizational principle Includes supplementary material: sn.pub/extras

Autorentext

Winfried Bruns has contributed numerous articles to homological and combinatorial commutative algebra. The book Cohen-Macaulay Rings he co-wrote with J. Herzog has become a standard reference. His work in discrete convex geometry is presented in the book Polytopes, Rings and K-Theory co-authored with J. Gubeladze, and in the software package Normaliz. Aldo Conca has written over seventy papers in commutative algebra. His main contributions are related to determinantal rings, Gröbner degenerations, Koszul and quadratic algebras, and Koszul homology. More recently, he has been involved in projects where commutative algebra is applied to "real world" problems. Claudiu Raicu has contributed to the study of homological invariants in commutative algebra and algebraic geometry, with an emphasis on problems involving symmetries coming from a group action. In the case of determinantal varieties and schemes, his work includes explicit calculations of a number of invariants such as local cohomology groups, Lyubeznik numbers, Hodge ideals, Ext modules and asymptotic regularity. Matteo Varbaro has contributed to the study of various topics in commutative algebra. His contributions include results on Gröbner deformations, determinantal objects, local cohomology, combinatorial commutative algebra, F-singularities, Castelnuovo-Mumford regularity.

Klappentext

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.

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.