Progress in Commutative Algebra 2 (eBook)

Preface 6
A Guide to Closure Operations in Commutative Algebra 12
1 Introduction 12
2 What Is a Closure Operation? 13
2.1 The Basics 13
2.2 Not-quite-closure Operations 17
3 Constructing Closure Operations 18
3.1 Standard Constructions 18
3.2 Common Closures as Iterations of Standard Constructions 20
4 Properties of Closures 21
4.1 Star-, Semi-prime, and Prime Operations 21
4.2 Closures Defined by Properties of (Generic) Forcing Algebras 27
4.3 Persistence 28
4.4 Axioms Related to the Homological Conjectures 29
4.5 Tight Closure and Its Imitators 31
4.6 (Homogeneous) Equational Closures and Localization 32
5 Reductions, Special Parts of Closures, Spreads, and Cores 33
5.1 Nakayama Closures and Reductions 33
5.2 Special Parts of Closures 34
6 Classes of Rings Defined by Closed Ideals 36
6.1 When Is the Zero Ideal Closed? 37
6.2 When Are 0 and Principal Ideals Generated by Non-zerodivisors Closed? 37
6.3 When Are Parameter Ideals Closed (Where R Is Local)? 38
6.4 When Is Every Ideal Closed? 39
7 Closure Operations on (Sub)modules 40
7.1 Torsion Theories 42
A Survey of Test Ideals 50
1 Introduction 50
2 Characteristic p Preliminaries 52
2.1 The Frobenius Endomorphism 52
2.2 F-purity 53
3 The Test Ideal 55
3.1 Test Ideals of Map-pairs 55
3.2 Test Ideals of Rings 58
3.3 Test Ideals in Gorenstein Local Rings 59
4 Connections with Algebraic Geometry 61
4.1 Characteristic 0 Preliminaries 61
4.2 Reduction to Characteristic p > 0 and Multiplier Ideals
4.3 Multiplier Ideals of Pairs 65
4.4 Multiplier Ideals vs. Test Ideals of Divisor Pairs 67
5 Tight Closure and Applications of Test Ideals 68
5.1 The Briançon-Skoda Theorem 72
5.2 Tight Closure for Modules and Test Elements 72
6 Test Ideals for Pairs (R, at) and Applications 74
6.1 Initial Definitions of at -test Ideals 74
6.2 at -tight Closure 76
6.3 Applications 77
7 Generalizations of Pairs: Algebras of Maps 79
8 Other Measures of Singularities in Characteristic p 82
8.1 F-rationality 82
8.2 F-injectivity 83
8.3 F-signature and F-splitting Ratio 84
8.4 Hilbert-Kunz(-Monsky) Multiplicity 86
8.5 F-ideals, F-stable Submodules, and F-pure Centers 89
A Canonical Modules and Duality 91
A.1 Canonical Modules, Cohen-Macaulay and Gorenstein Rings 91
A.2 Duality 92
B Divisors 94
C Glossary and Diagrams on Types of Singularities 96
C.1 Glossary of Terms 97
Finite-dimensional Vector Spaces with Frobenius Action 112
1 Introduction 112
2 A Noncommutative Principal Ideal Domain 113
3 Ideal Theory and Divisibility in Noncommutative PIDs 115
3.1 Examples in K{F} 118
4 Matrix Transformations over Noncommutative PIDs 120
5 Module Theory over Noncommutative PIDs 122
6 Computing the Invariant Factors 125
6.1 Injective Frobenius Actions on Finite Dimensional Vector Spaces over a Perfect Field 129
7 The Antinilpotent Case 132
Finiteness and Homological Conditions in Commutative Group Rings 140
1 Introduction 140
2 Finiteness Conditions 141
3 Homological Dimensions and Regularity 144
4 Zero Divisor Controlling Conditions 147
Regular Pullbacks 156
1 Introduction 156
2 Some Background 158
3 Pullbacks of Noetherian Rings 162
4 Pullbacks of Prüfer Rings 164
5 Pullbacks of Coherent Rings 167
6 The n -generator Property in Pullbacks 170
7 Factorization in Pullbacks 176
Noetherian Rings without Finite Normalization 182
1 Introduction 182
2 Normalization and Completion 185
3 Examples between DVRs 187
4 Examples Birationally Dominating a Local Ring 196
5 A Geometric Example 200
6 Strongly Twisted Subrings of Local Noetherian Domains 201
Krull Dimension of Polynomial and Power Series Rings 216
1 Introduction 216
2 A Key Property of R[x] 218
3 The Main Theorem 219
4 Additional Applications 223
5 The Dimension of Power Series Rings 228
The Projective Line over the Integers 232
1 Introduction 232
2 Definitions and Background 233
3 The Coefficient Subset and Radical Elements of Proj (Z[h,k]) 237
4 The Conjecture for Proj (Z[h, k]) and Previous Partial Results 240
5 New Results Supporting the Conjecture 243
6 Summary and Questions 249
On Zero Divisor Graphs 252
1 Introduction 252
2 Survey of Past Research on Zero Divisor Graphs 254
2.1 Beck’s Zero Divisor Graph 254
2.2 Anderson and Livingston’s Zero Divisor Graph 255
2.3 Mulay’s Zero Divisor Graph 257
2.4 Other Zero Divisor Graphs 258
3 Star Graphs 261
4 Graph Homomorphisms and Graphs Associated to Modules 281
5 Cliques 284
6 Girth and Cut Vertices 293
6.1 Girth 293
6.2 Cut Vertices 297
7 Chromatic Numbers and Clique Numbers 299
7.1 Chromatic/Clique Number 1 300
7.2 Chromatic/Clique Number 2 301
7.3 Chromatic/Clique Number 3 301
A Tables for Example 3.14 304
B Graph Theory 306
A Closer Look at Non-unique Factorization via Atomic Decay and Strong Atoms 312
1 Introduction 312
2 Strong Atoms and Prime Ideals 314
3 Atomic Decay in the Ring of Integers of an Algebraic Number Field 316
4 The Fundamental Example of the Failure of Unique Factorization: Z[v-5] 320
5 A More Striking Example 322
6 Concluding Remarks and Questions 325