Polytopes, Rings, and K-Theory (eBook)

Preface 6
Contents 9
Cones, monoids, and triangulations 12
Polytopes, cones, and complexes 13
A Polyhedra and their faces 13
B Finite generation of cones 20
C Finite generation of polyhedra 25
D Polyhedral complexes 31
E Subdivisions and triangulations 35
F Regular subdivisions 41
G Rationality and integrality 49
Exercises 54
Notes 57
Affine monoids and their Hilbert bases 59
A Affine monoids 59
B Normal affine monoids 68
C Generating normal affine monoids 77
D Normality and unimodular covering 87
Exercises 95
Notes 98
Multiples of lattice polytopes 100
A Knudsen–Mumford triangulations 100
B Unimodular triangulations of multiples of polytopes 106
C Unimodular covers of multiples of polytopes 112
Exercises 126
Notes 128
Affine monoid algebras 129
Monoid algebras 130
A Graded rings 130
B Monoid algebras 136
C Representations of monoid algebras 140
D Monomial prime and radical ideals 144
E Normality 147
F Divisorial ideals and the class group 152
G The Picard group and seminormality 160
Exercises 167
Notes 169
Isomorphisms and automorphisms 171
A Linear algebraic groups 171
B Invariants of diagonalizable groups 178
C The isomorphism theorem 181
D Automorphisms 189
Exercises 201
Notes 204
Homological properties andHilbert functions 205
A Cohen–Macaulay rings 205
B Graded homological algebra 212
C The canonical module 219
D Hilbert functions 224
E Applications to enumerative combinatorics 233
F Divisorial linear algebra 243
Exercises 252
Notes 254
Gröbner bases, triangulations, and Koszul algebras 256
A Gröbner bases and initial ideals 256
B Initial ideals of toric ideals 261
C Toric ideals and triangulations 272
D Multiples of lattice polytopes 281
Exercises 286
Notes 287
K -theory 289
Projectivemodules overmonoid rings 290
A Projectivemodules 290
B The main theoremand the plan of the proof 292
C Projectivemodules over polynomial rings 295
D Reduction to the interior 300
E Graded "Weierstrass preparation” 301
F Pyramidal descent 302
G How to shrink a polytope 308
H Converse results 310
I Generalizations 312
Exercises 324
Notes 327
Bass–Whitehead groups ofmonoid rings 329
A The functors K1 and K2 329
B The nontriviality of SK1(R[M]) 335
C Further results: a survey 347
Exercises 353
Notes 355
Varieties 356
A Vector bundles, coherent sheaves, and Grothendieck groups 357
B Toric varieties 365
C Chow groups of toric varieties 376
D Intersection theory 381
E Chow cohomology of toric varieties 394
F Toric varieties with huge Grothendieck group 405
G The equivariant Serre problem for abelian groups 414
Exercises 426
References 430
Notation 445
Index 451