Integral closure of ideals, rings and modules

Integral closure has played a role in number theory and algebraic geometry since the nineteenth century, but a modern formulation of the concept for ideals perhaps began with the work of Krull and Zariski in the 1930s. It has developed into a tool for the analysis of many algebraic and geometric problems. This book collects together the central notions of integral closure and presents a unified treatment. Techniques and topics covered include: behavior of the Noetherian property under integral closure, analytically unramified rings, the conductor, field separability, valuations, Rees algebras, Rees valuations, reductions, multiplicity, mixed multiplicity, joint reductions, the Briançon-Skoda theorem, Zariski's theory of integrally closed ideals in two-dimensional regular local rings, computational aspects, adjoints of ideals and normal homomorphisms. With many worked examples and exercises, this book will provide graduate students and researchers in commutative algebra or ring theory with an approachable introduction leading into the current literature.

Contents	p. v
Table of basic properties	p. ix
Notation and basic definitions	p. xi
Preface	p. xiii
1 What is integral closure of ideals?	p. 1
1.1 Basic properties	p. 2
1.2 Integral closure via reductions	p. 5
1.3 Integral closure of an ideal is an ideal	p. 6
1.4 Monomial ideals	p. 9
1.5 Integral closure of rings	p. 13
1.6 How integral closure arises	p. 14
1.7 Dedekind-Mertens formula	p. 17
1.8 Exercises	p. 20
2 Integral closure of rings	p. 23
2.1 Basic facts	p. 23
2.2 Lying-Over, Incomparability, Going-Up, Going-Down	p. 30
2.3 Integral closure and grading	p. 34
2.4 Rings of homomorphisms of ideals	p. 39
2.5 Exercises	p. 42
3 Separability	p. 47
3.1 Algebraic separability	p. 47
3.2 General separability	p. 48
3.3 Relative algebraic closure	p. 52
3.4 Exercises	p. 54
4 Noetherian rings	p. 56
4.1 Principal ideals	p. 56
4.2 Normalization theorems	p. 57
4.3 Complete rings	p. 60
4.4 Jacobian ideals	p. 63
4.5 Serre's conditions	p. 70
4.6 Affine and Z-algebras	p. 73
4.7 Absolute integral closure	p. 77
4.8 Finite Lying-Over and height	p. 79
4.9 Dimension one	p. 83
4.10 Krull domains	p. 85
4.11 Exercises	p. 89
5 Rees algebras	p. 93
5.1 Rees algebra constructions	p. 93
5.2 Integral closure of Rees algebras	p. 95
5.3 Integral closure of powers of an ideal	p. 97
5.4 Powers and formal equidimensionality	p. 100
5.5 Defining equations of Rees algebras	p. 104
5.6 Blowing up	p. 108
5.7 Exercises	p. 109
6 Valuations	p. 113
6.1 Valuations	p. 113
6.2 Value groups and valuation rings	p. 115
6.3 Existence of valuation rings	p. 117
6.4 More properties of valuation rings	p. 119
6.5 Valuation rings and completion	p. 121
6.6 Some invariants	p. 124
6.7 Examples of valuations	p. 130
6.8 Valuations and the integral closure of ideals	p. 133
6.9 The asymptotic Samuel function	p. 138
6.10 Exercises	p. 139
7 Derivations	p. 143
7.1 Analytic approach	p. 143
7.2 Derivations and differentials	p. 147
7.3 Exercises	p. 149
8 Reductions	p. 150
8.1 Basic properties and examples	p. 150
8.2 Connections with Rees algebras	p. 154
8.3 Minimal reductions	p. 155
8.4 Reducing to infinite residue fields	p. 159
8.5 Superficial elements	p. 160
8.6 Superficial sequences and reductions	p. 165
8.7 Non-local rings	p. 169
8.8 Sally's theorem on extensions	p. 171
8.9 Exercises	p. 173
9 Analytically unramified rings	p. 177
9.1 Rees's characterization	p. 178
9.2 Module-finite integral closures	p. 180
9.3 Divisorial valuations	p. 182
9.4 Exercises	p. 185
10 Rees valuations	p. 187
10.1 Uniqueness of Rees valuations	p. 187
10.2 A construction of Rees valuations	p. 191
10.3 Examples	p. 196
10.4 Properties of Rees valuations	p. 201
10.5 Rational powers of ideals	p. 205
10.6 Exercises	p. 208
11 Multiplicity and integral closure	p. 212
11.1 Hilbert-Samuel polynomials	p. 212
11.2 Multiplicity	p. 217
11.3 Rees's theorem	p. 222
11.4 Equimultiple families of ideals	p. 225
11.5 Exercises	p. 232
12 The conductor	p. 234
12.1 A classical formula	p. 235
12.2 One-dimensional rings	p. 235
12.3 The Lipman-Sathaye theorem	p. 237
12.4 Exercises	p. 242
13 The Briancon-Skoda Theorem	p. 244
13.1 Tight closure	p. 245
13.2 Briancon-Skoda via tight closure	p. 248
13.3 The Lipman-Sathaye version	p. 250
13.4 General version	p. 253
13.5 Exercises	p. 256
14 Two-dimensional regular local rings	p. 257
14.1 Full ideals	p. 258
14.2 Quadratic transformations	p. 263
14.3 The transform of an ideal	p. 266
14.4 Zariski's theorems	p. 268
14.5 A formula of Hoskin and Deligne	p. 274
14.6 Simple integrally closed ideals	p. 277
14.7 Exercises	p. 279
15 Computing integral closure	p. 281
15.1 Method of Stolzenberg	p. 282
15.2 Some computations	p. 286
15.3 General algorithms	p. 292
15.4 Monomial ideals	p. 295
15.5 Exercises	p. 297
16 Integral dependence of modules	p. 302
16.1 Definitions	p. 302
16.2 Using symmetric algebras	p. 304
16.3 Using exterior algebras	p. 307
16.4 Properties of integral closure of modules	p. 309
16.5 Buchsbaum-Rim multiplicity	p. 313
16.6 Height sensitivity of Koszul complexes	p. 319
16.7 Absolute integral closures	p. 321
16.8 Complexes acyclic up to integral closure	p. 325
16.9 Exercises	p. 327
17 Joint reductions	p. 331
17.1 Definition of joint reductions	p. 331
17.2 Superficial elements	p. 333
17.3 Existence of joint reductions	p. 335
17.4 Mixed multiplicities	p. 338
17.5 More manipulations of mixed multiplicities	p. 344
17.6 Converse of Rees's multiplicity theorem	p. 348
17.7 Minkowski inequality	p. 350
17.8 The Rees-Sally formulation and the core	p. 353
17.9 Exercises	p. 358
18 Adjoints of ideals	p. 360
18.1 Basic facts about adjoints	p. 360
18.2 Adjoints and the Briancon-Skoda Theorem	p. 362
18.3 Background for computation of adjoints	p. 364
18.4 Adjoints of monomial ideals	p. 366
18.5 Adjoints in two-dimensional regular rings	p. 369
18.6 Mapping cones	p. 372
18.7 Analogs of adjoint ideals	p. 375
18.8 Exercises	p. 376
19 Normal homomorphisms	p. 378
19.1 Normal homomorphisms	p. 379
19.2 Locally analytically unramified rings	p. 381
19.3 Inductive limits of normal rings	p. 383
19.4 Base change and normal rings	p. 384
19.5 Integral closure and normal maps	p. 388
19.6 Exercises	p. 390
Appendix A Some background material	p. 392
A.1 Some forms of Prime Avoidance	p. 392
A.2 Caratheodory's theorem	p. 392
A.3 Grading	p. 393
A.4 Complexes	p. 394
A.5 Macaulay representation of numbers	p. 396
Appendix B Height and dimension formulas	p. 397
B.1 Going-Down, Lying-Over, flatness	p. 397
B.2 Dimension and height inequalities	p. 398
B.3 Dimension formula	p. 399
B.4 Formal equidimensionality	p. 401
B.5 Dimension Formula	p. 403
References	p. 405
Index	p. 422

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