Integral Closure: Rees Algebras, Multiplicities, Algorithms

Contents......Page 8
Preface......Page 6
Introduction......Page 12
1 Numerical Invariants of a Rees Algebra......Page 28
1.1 Equations of a Rees Algebra......Page 30
1.1.1 The Rees Algebra of an Ideal......Page 31
1.1.2 Dimension of Symmetric and Rees Algebras of Modules......Page 38
1.2 Rees Algebras and Reductions......Page 42
1.2.1 Basic Properties of Reductions......Page 44
1.2.2 Integrally Closed Ideals and Normal Ideals......Page 50
1.3 Special Fiber and Analytic Spread......Page 56
1.3.1 Special Fiber and Noether Normalization......Page 57
1.3.2 Explicit Reduction Numbers......Page 58
1.3.3 Analytic Spread and Codimension......Page 61
1.4 Reduction Numbers of Ideals......Page 65
1.5 Determinants and Ranks of Modules......Page 76
1.5.1 Cayley-Hamilton Theorem......Page 77
1.5.2 The Big Rank of a Module......Page 84
1.6.1 Arithmetic Degree......Page 86
1.6.2 Global Bounds of Reduction Numbers......Page 91
1.7 Intertwining Algebras......Page 94
1.8 Briançon-Skoda Bounds......Page 101
1.9 Exercises......Page 103
2 Hilbert Functions and Multiplicities......Page 107
2.1.1 Structures Associated to Rees Algebras......Page 110
2.1.2 Bounding Hilbert Functions......Page 116
2.2 Maximal Hilbert Functions......Page 128
2.2.1 The Eakin-Sathaye Theorem......Page 129
2.2.2 Hilbert Functions of Primary Ideals......Page 131
2.3.1 Classical Degrees......Page 140
2.3.2 Generalized Multiplicities of Graded Modules......Page 146
2.4 Cohomological Degrees......Page 150
2.4.1 Homological Degree......Page 151
2.4.2 General Properties of Degs......Page 158
2.5 Finiteness of Hilbert Functions......Page 172
2.6.1 Estimating Number of Generators with Multiplicities......Page 175
2.6.2 Number of Generators and the Socle......Page 180
2.7 Multiplicities and Reduction Numbers......Page 186
2.7.1 The Modulo Dimension One Technique......Page 188
2.7.2 Special Fibers......Page 189
2.7.3 Ideals of Dimension One and Two......Page 198
2.8 Exercises......Page 210
3 Depth and Cohomology of Rees Algebras......Page 214
3.1.1 Systems of Parameters and Hypersurface Sections......Page 215
3.1.2 Passing Cohen-Macaulayness Around......Page 219
3.2 Cohen-Macaulayness of Proj (R) and Cohomology......Page 224
3.2.1 Castelnuovo-Mumford Regularity and a-invariants......Page 227
3.2.2 Vanishing of Cohomology......Page 230
3.3 Reduction Number and Cohen-Macaulayness......Page 234
3.4.1 Detecting (S[sub(k)])......Page 241
3.4.2 R[sub(k)]-Conditions on Rees Algebras......Page 243
3.5 Exercises......Page 244
4 Divisors of a Rees Algebra......Page 246
4.1 Divisors of an Algebra......Page 248
4.2 Divisor Class Group......Page 255
4.3 The Expected Canonical Module......Page 263
4.4 The Fundamental Divisor......Page 269
4.5 Cohen-Macaulay Divisors and Reduction Numbers......Page 277
4.6 Exercises......Page 278
5 Koszul Homology......Page 280
5.1 Koszul Complexes of Ideals and Modules......Page 282
5.2 Module Structure of Koszul Homology......Page 289
5.3 Linkage and Residual Intersections......Page 298
5.4 Approximation Complexes......Page 301
5.5 Ideals with Good Reductions......Page 309
5.6 Exercises......Page 315
6 Integral Closure of Algebras......Page 316
6.1.1 Noether Normalization......Page 319
6.1.2 Canonical Module and S[sub(2)]-ification......Page 326
6.2.1 The Jacobian Ideal......Page 329
6.2.2 R[sub(1)]-ification......Page 332
6.2.3 The Integral Closure of Subrings Defined by Graphs......Page 335
6.3 Divisorial Extensions of an Affine Algebra......Page 338
6.3.1 Divisorial Extensions of Gorenstein Rings......Page 339
6.3.2 Non-Homogeneous Algebras......Page 342
6.4.1 Chern Coefficients......Page 346
6.4.2 Bounding Tracking Numbers......Page 352
6.5 Embedding Dimension of the Integral Closure......Page 357
6.5.1 Cohen-Macaulay Integral Closure......Page 359
6.5.2 Small Singularities......Page 366
6.6 Arithmetic Affine Algebras......Page 374
6.7 Exercises......Page 378
7 Integral Closure and Normalization of Ideals......Page 380
7.1 Hilbert Functions and Integral Closure......Page 382
7.2 Monomial Ideals......Page 390
7.3 Multiplicities and Volumes......Page 395
7.4 Normalization of an Ideal......Page 409
7.5 Algebras of Symbolic Powers......Page 418
7.6 Exercises......Page 420
8 Integral Closure of Modules......Page 422
8.1 Dimensions of Rees Algebras and of their Fibers......Page 425
8.2 Rees Integrality Criteria......Page 429
8.3 Reduction Numbers of Modules......Page 437
8.3.1 Reduction Number of a Module......Page 441
8.3.2 Extended Degree for the Buchsbaum-Rim Multiplicity......Page 446
8.4 Divisors of Modules and Integral Closure......Page 447
8.4.1 Order Ideal of a Module......Page 448
8.4.2 Determinantal Ideals and Reductions......Page 454
8.4.3 m-full Modules......Page 459
8.5 Normality of Algebras of Linear Type......Page 462
8.5.1 Complete Intersection Modules......Page 463
8.5.2 Symbolic Powers and Normal Modules......Page 465
8.5.3 Complete Modules and Finite Projective Dimension......Page 474
8.5.4 Determinantal Ideals of Symmetric Powers......Page 476
8.6 Bourbaki Ideals and Rees Algebras......Page 478
8.7 Normalization of Modules......Page 482
8.8 Exercises......Page 487
9.1 Module Operations......Page 489
9.2 Integral Closure of an Algebra......Page 491
9.3 Integral Closure of an Ideal......Page 493
9.4 Integral Closure of a Module......Page 498
9.5 Exercises......Page 503
References......Page 504
Notation and Terminology......Page 520
B......Page 522
E......Page 523
L......Page 524
R......Page 525
Z......Page 526