Multiplicative Invariant Theory (eBook)

Preface 6
Contents 8
Introduction 11
The Setting 11
Multiplicative and Polynomial Invariants 12
Special Features of Multiplicative Actions 13
Origins and Uses of Multiplicative Invariants 14
Overview of the Contents 15
Notations and Conventions 18
1 Groups Acting on Lattices 21
1.1 Introduction 21
1.2 G-Lattices 21
1.3 Examples 22
1.4 Standard Lattice Constructions 24
1.5 Indecomposable Lattices 26
1.6 Conditioning the Lattice 28
1.7 Reflections and Generalized Reflections 29
1.8 Lattices Associated with Root Systems 32
1.9 The Root System Associated with a Faithful G-Lattice 35
1.10 Finite Subgroups of GLn(Z) 36
2 Permutation Lattices and Flasque Equivalence 40
2.1 Introduction 40
2.2 Permutation Lattices 40
2.3 Stable Permutation Equivalence 41
2.4 Permutation Projective Lattices 42
2.5 Hi-trivial, Flasque and Coflasque Lattices 43
2.6 Flasque and Coflasque Resolutions 44
2.7 Flasque Equivalence 45
2.8 Quasi-permutation Lattices and Monomial Lattices 46
2.9 An Invariant for Flasque Equivalence 47
2.10 Overview of Lattice Types 49
2.11 Restriction to the Sylow Normalizer 50
2.12 Some Sn - Lattices 51
3 Multiplicative Actions 58
3.1 Introduction 58
3.2 The Group Algebra of a G-Lattice 58
3.3 Reduction to Finite Groups, Z -structure, and Finite Generation 59
3.4 Units and Semigroup Algebras 60
3.5 Examples 62
3.6 Multiplicative Invariants ofWeight Lattices 68
3.7 Passage to an Effective Lattice 70
3.8 Twisted Multiplicative Actions 71
3.9 Hopf Structure 73
3.10 Torus Invariants 74
4 Class Group 75
4.1 Introduction 75
4.2 Some Examples 76
4.3 Krull Domains and Class Groups 76
4.4 Samuel’s Exact Sequence 78
4.5 Generalized Reflections on Rings 79
4.6 Proof of Theorem 4.1.1 81
5 Picard Group 82
5.1 Introduction 82
5.2 Invertible Modules 83
5.3 The Skew Group Ring 84
5.4 The Trace Map 85
5.5 The Kernel of the Map Pic(RG) . Pic(R) 86
5.6 The Case of Multiplicative Actions 88
6 Multiplicative Invariants of Reflection Groups 90
6.1 Introduction 90
6.2 Proof of Theorem 6.1.1 91
6.3 Computing the Ring of Invariants 92
6.4 SAGBI Bases 96
7 Regularity 100
7.1 Introduction 100
7.2 Projectivity over Invariants 101
7.3 Linearization by the Slice Method 102
7.4 Proof of Theorem 7.1.1 104
7.5 Regularity at the Identity 105
8 The Cohen-Macaulay Property 107
8.1 Introduction 107
8.2 Height and Grade 108
8.3 Local Cohomology 109
8.4 Cohen-Macaulay Modules and Rings 110
8.5 The Cohen-Macaulay Property for Invariant Rings 111
8.6 The Ellingsrud-Skjelbred Spectral Sequences 114
8.7 Annihilators of Cohomology Classes 116
8.8 The Restriction Map for Cohen-Macaulay Invariants 118
8.9 The Case of Multiplicative Invariants 120
8.10 Proof of Theorem 8.1.1 122
8.11 Examples 125
9 Multiplicative Invariant Fields 128
9.1 Introduction 128
9.2 Stable Isomorphism 131
9.3 Retract Rationality 131
9.4 The “No-name Lemma" 134
9.5 Function Fields of Algebraic Tori 136
9.6 Some Rationality Results for Multiplicative Invariant Fields 141
9.7 Some Concepts from Algebraic Geometry 144
9.8 The Field of Matrix Invariants as a Multiplicative Invariant Field 145
10 Problems 152
10.1 The Cohen-Macaulay Problem 152
10.2 Semigroup Algebras 154
10.3 Computational Issues 157
10.4 Essential Dimension Estimates 158
10.5 Rationality Problems 162
References 164
Index 175