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Cover: 9781441926173 | Polytopes, Rings, and K-Theory | Joseph Gubeladze (u. a.) | Buch | XIV
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Polytopes, Rings, and K-Theory
Taschenbuch von Joseph Gubeladze (u. a.)
Sprache: Englisch

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Beschreibung
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.
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.
Zusammenfassung

This book examines interactions of polyhedral discrete geometry and algebra. What makes this book different from others is the presentation of several central results in all three areas of the exposition - discrete geometry, commutative algebra, and K-theory. The only prerequisite for the reader is a background in algebra, and the basics of polyhedral geometry have been included in Chapter 1.

The text will be of interest to graduate students and mathematicians. Included are numerous exercises, historical background, and notes throughout the chapters.

Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Springer Monographs in Mathematics
Inhalt: xiv
461 S.
52 s/w Illustr.
461 p. 52 illus.
ISBN-13: 9781441926173
ISBN-10: 1441926178
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Gubeladze, Joseph
Bruns, Winfried
Auflage: Softcover reprint of hardcover 1st ed. 2009
Hersteller: Springer New York
Springer US, New York, N.Y.
Springer Monographs in Mathematics
Maße: 235 x 155 x 26 mm
Von/Mit: Joseph Gubeladze (u. a.)
Erscheinungsdatum: 06.12.2010
Gewicht: 0,715 kg
Artikel-ID: 107152512
Zusammenfassung

This book examines interactions of polyhedral discrete geometry and algebra. What makes this book different from others is the presentation of several central results in all three areas of the exposition - discrete geometry, commutative algebra, and K-theory. The only prerequisite for the reader is a background in algebra, and the basics of polyhedral geometry have been included in Chapter 1.

The text will be of interest to graduate students and mathematicians. Included are numerous exercises, historical background, and notes throughout the chapters.

Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Springer Monographs in Mathematics
Inhalt: xiv
461 S.
52 s/w Illustr.
461 p. 52 illus.
ISBN-13: 9781441926173
ISBN-10: 1441926178
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Gubeladze, Joseph
Bruns, Winfried
Auflage: Softcover reprint of hardcover 1st ed. 2009
Hersteller: Springer New York
Springer US, New York, N.Y.
Springer Monographs in Mathematics
Maße: 235 x 155 x 26 mm
Von/Mit: Joseph Gubeladze (u. a.)
Erscheinungsdatum: 06.12.2010
Gewicht: 0,715 kg
Artikel-ID: 107152512
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