Contents
1. Abstract vii
2. Introduction 1
Chapter 0. Preliminaries 9
3. Standard Notation and Terminology 9
4. Q-Localization and Minimal Primes 10
5. Ideal-adic Completions 12
6. Primary Components (terminology from abelian groups) 15
7. Elimination of finite-length summands 17
8. Basic Finiteness Conditions 20
9. General Cancellation and Direct Summands in Dimension 1 22
Chapter 1. Dedekind-like Rings 25
10. Definition and Characterizations 25
11. Maximal Ideals: Residue Inclusions, Localizations, Completions 27
12. Surjective Pullback Squares 32
13. Homomorphic Images: Local versus Complete Local 34
Chapter 2. Wildness 39
14. Global Dichotomy, Global Wildness 39
Chapter 3. Structure of a Genus 45
15. Consistent (Torsionfree) Ranks 45
16. Indecomposable A-modules; Connections Graph 49
Chapter 4. Substitute for Conductor Squares 61
17. Residue Inclusions of Separated Modules 61
18. Separated Covers 66
Chapter 5. Isomorphism Classes in a Genus, Idele Group Action 75
19. Restricted Genus and Separated Covers 75
20. Wiegand Lifting Theorem 77
21. Residue Endo-ideles on T-modules 82
22. Residue Unit Ideles 87
23. Stabilizers: Basic properties 92
24. Equality Stabilizers 96
Chapter 6. Web of Class Groups 113
25. Genus Class Group, £-map, Mayer-Vietoris Sequence 113
26. Cancellation, Restricted Genus, and Ideles 120
27. Super Mayer-Vietoris and faithful modules 123
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