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 initialideals 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
References 429
Notation 445
Index 451
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