Polytopes, Rings, and K-Theory

ISBN-13: 9781441926173 / Angielski / Miękka / 2010 / 461 str.

Springer

Opis

For every mathematician, ring theory and K-theory are intimately connected: al- braic K-theory is largely the K-theory of rings. At ?rst sight, polytopes, by their very nature, must appear alien to surveyors of this heartland of algebra. But in the presence of a discrete structure, polytopes de?ne a?ne monoids, and, in their turn, a?ne monoids give rise to monoid algebras. Teir spectra are the building blocks of toric varieties, an area that has developed rapidly in the last four decades. From a purely systematic viewpoint, "monoids" should therefore replace "po- topes" in the title of the book. However, such a change would conceal the geometric ?avor that we have tried to preserve through all chapters. Before delving into a description of the contents we would like to mention three general features of the book: (?) the exhibiting of interactions of convex geometry, ring theory, and K-theory is not the only goal; we present some of the central results in each of these ?elds; (?) the exposition is of constructive (i. e., algorithmic) nature at many places throughout the text-there is no doubt that one of the driving forces behind the current popularity of combinatorial geometry is the quest for visualization and computation; (?) despite the large amount of information from various ?elds, we have strived to keep the polytopal perspective as the major organizational principle.

Recenzje

From the reviews:

"The book deals with the convex geometry of the polyhedral cones CM and related topics ... . With an extensive list of references together with historical notes at the end of each chapter, this book will serve as a welcome comprehensive monograph for research on these rich and fascinating subjects." (T. Oda, Mathematical Reviews, Issue 2010 d)

"Interactions between convex geometry, ring theory, K-theory, combinatorial geometry, toric geometry, combinatorics in commutative algebra, which are presented in this book together with ... central results in each of the above fields. ... All the chapters contain many useful exercises ... . a good part of the book is not covered by any other book and so especially people from commutative algebra should have it." (Dorin-Mihail Popescu, Zentralblatt MATH, Vol. 1168, 2009)

"Polytopes, Rings, and K-Theory weighs in at over 400 pages and ten chapters split into three main parts, culminating in the aforementioned K-theory in the given context. ... There are exercises galore in the book ... . All in all, Polytopes, Rings, and K-Theory is an accessible and well-written book on an interesting and important subject ... . It should be quite a success." (Michael Berg, The Mathematical Association of America, December, 2009)

Spis treści

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.

Blurb

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.