Polytopes, rings, and K-theory / by Winfried Bruns, Joseph Gubeladze.

Format:

Book

Author/Creator:

Bruns, Winfried, 1946-

Contributor:

Gubeladze, Joseph.

Series:

Springer monographs in mathematics

Language:

English

Subjects (All):

Polytopes.

K-theory.

Rings (Algebra).

Physical Description:

xiii, 461 pages : illustrations ; 24 cm.

Place of Publication:

New York ; London : Springer, 2009.

Summary:

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 Universitat Osnabruck. Joseph Gubeladze is Professor of Mathematics at San Francisco State University.

Contents:

Part I Cones, monoids, and triangulations

1 Polytopes, cones, and complexes 3

1.A Polyhedra and their faces 3

1.B Finite generation of cones 10

1.C Finite generation of polyhedra 15

1.D Polyhedral complexes 21

1.E Subdivisions and triangulations 25

1.F Regular subdivisions 31

1.G Rationality and integrality 39

Exercises 44

Notes 47

2 Affine monoids and their Hilbert bases 49

2.A Affine monoids 49

2.B Normal affine monoids 58

2.C Generating normal affine monoids 67

2.D Normality and unimodular covering 77

Exercises 85

Notes 88

3 Multiples of lattice polytopes 91

3.A Knudsen-Mumford triangulations 91

3.B Unimodular triangulations of multiples of polytopes 97

3.C Unimodular covers of multiples of polytopes 103

Exercises 117

Notes 119

Part II Affine monoid algebras

4 Monoid algebras 123

4.A Graded rings 123

4.B Monoid algebras 129

4.C Representations of monoid algebras 133

4.D Monomial prime and radical ideals 137

4.E Normality 140

4.F Divisorial ideals and the class group 145

4.G The Picard group and seminormality 153

Exercises 160

Notes 162

5 Isomorphisms and automorphisms 165

5.A Linear algebraic groups 165

5.B Invariants of diagonalizable groups 172

5.C The isomorphism theorem 175

5.D Automorphisms 183

Exercises 195

Notes 198

6 Homological properties and Hilbert functions 199

6.A Cohen-Macaulay rings 199

6.B Graded homological algebra 206

6.C The canonical module 213

6.D Hilbert functions 218

6.E Applications to enumerative combinatorics 227

6.F Divisorial linear algebra 237

Exercises 246

Notes 248

7 Grobner bases, triangulations, and Koszul algebras 251

7.A Grobner bases and initial ideals 251

7.B Initial ideals of toric ideals 256

7.C Toric ideals and triangulations 267

7.D Multiples of lattice polytopes 276

Exercises 281

Notes 282

Part III K-theory

8 Projective modules over monoid rings 287

8.A Projective modules 287

8.B The main theorem and the plan of the proof 289

8.C Projective modules over polynomial rings 292

8.D Reduction to the interior 297

8.E Graded "Weierstrass preparation" 298

8.F Pyramidal descent 299

8.G How to shrink a polytope 305

8.H Converse results 307

8.I Generalizations 309

Exercises 321

Notes 324

9 Bass-Whitehead groups of monoid rings 327

9.A The functors K₁ and K₂ 327

9.B The nontriviality of SK₁ (R[M]) 333

9.C Further results: a survey 345

Exercises 351

Notes 353

10 Varieties 355

10.A Vector bundles, coherent sheaves, and Grothendieck groups 356

10.B Toric varieties 364

10.C Chow groups of toric varieties 375

10.D Intersection theory 380

10.E Chow cohomology of toric varieties 393

10.F Toric varieties with huge Grothendieck group 404

10.G The equivariant Serre problem for abelian groups 413

Exercises 425.

Notes:

Includes bibliographical references (pages [429]-443) and index.

ISBN:

9780387763552

0387763554

OCLC:

416289393