Polytopes, Rings, and K-Theory

Winfried Bruns, Joseph Gubeladze

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English

Springer-Verlag New York Inc.
27 May 2009

Discrete mathematics; Algebra; Geometry; Algebraic geometry; Algebraic topology

Series: Springer Monographs in Mathematics

"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."

By: Winfried Bruns, Joseph Gubeladze
Imprint: Springer-Verlag New York Inc.
Country of Publication: United States
Edition: 2009 ed.
Dimensions: Height: 235mm, Width: 155mm, Spine: 26mm
Weight: 1.850kg
ISBN: 9780387763552
ISBN 10: 0387763554
Series: Springer Monographs in Mathematics
Pages: 461
Publication Date: 27 May 2009
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Reviews for Polytopes, Rings, and K-Theory

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)