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Polytopes, Rings, and K-Theory Winfried Bruns

Polytopes, Rings, and K-Theory By Winfried Bruns

Polytopes, Rings, and K-Theory by Winfried Bruns


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Summary

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.

Polytopes, Rings, and K-Theory Summary

Polytopes, Rings, and K-Theory by Winfried Bruns

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.

Polytopes, Rings, and K-Theory Reviews

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)

Table of Contents

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.

Additional information

NLS9781441926173
9781441926173
1441926178
Polytopes, Rings, and K-Theory by Winfried Bruns
New
Paperback
Springer-Verlag New York Inc.
2010-12-06
461
N/A
Book picture is for illustrative purposes only, actual binding, cover or edition may vary.
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