Polytopes, Rings, and K-Theory (Springer Monographs in Mathematics)

Bruns, Winfried:

ISBN 10: 0387763554 / ISBN 13: 9780387763552

Editore: New York. Springer-Verlag., 2009

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Codice articolo 494910

Dati bibliografici

• Titolo: Polytopes, Rings, and K-Theory (Springer Monographs in Mathematics)
• Editore: New York. Springer-Verlag.
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 0387763554
• ISBN 13: 9780387763552
• Rilegatura: Karton Karton
• Condizione: Sehr gut
• Peso: 836 grammi
• Cataloghi dei venditori:
  • Mathematik

Descrizione articolo

Riassunto

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.

Dalla quarta di copertina

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.