Integral Domains Inside Noetherian Power Series Rings: Constructions and Examples (Mathematical Surveys and Monographs, 259)

Cover
Title page
Preface
Chapter 1. Introduction
Chapter 2. Tools
	2.1. Conventions and terminology
	2.2. Basic theorems
	2.3. Flatness
	Exercises
Chapter 3. More tools
	3.1. Introduction to ideal-adic completions
	3.2. Uncountable transcendence degree for a completion
	3.3. Power series pitfalls
	3.4. Basic results about completions
	3.5. Chains of prime ideals, fibers of maps
	3.6. Henselian rings
	3.7. Regularity and excellence
	Exercises
Chapter 4. First examples of the construction
	4.1. Universality
	4.2. Elementary examples
	4.3. Historical examples
	4.4. Prototypes
	Exercises
Chapter 5. The Inclusion Construction
	5.1. The Inclusion Construction and a picture
	5.2. Approximations for the Inclusion Construction
	5.3. Basic properties of the constructed domains
	Exercises
Chapter 6. Flatness and the Noetherian property
	6.1. The Noetherian Flatness Theorem
	6.2. Introduction to Insider Construction 10.7
	6.3. Nagata’s example
	6.4. Christel’s Example
	6.5. Further implications of the Noetherian Flatness Theorem
	Exercise
Chapter 7. The flat locus of an extension of polynomial rings
	7.1. Flatness criteria
	7.2. The Jacobian ideal and the smooth and flat loci
	7.3. Applications to polynomial extensions
	Exercises
Chapter 8. Excellent rings and formal fibers
	8.1. Background for excellent rings
	8.2. Nagata rings and excellence
	8.3. Henselian rings
	Exercises
Chapter 9. Height-one prime ideals and weak flatness
	9.1. The limit-intersecting condition
	9.2. Height-one primes in extensions of Krull domains
	9.3. Flatness for Inclusion Construction 5.3
	Exercises
Chapter 10. Insider Construction details
	10.1. Localized polynomial rings over special DVRs
	10.2. Describing the construction
	10.3. The non-flat locus of Insider Construction 10.7
	10.4. Preserving excellence with the Insider Construction
Chapter 11. Integral closure under extension to the completion
	11.1. Integral closure under ring extension
	11.2. Extending ideals to the completion
	11.3. Comments and questions
	Exercises
Chapter 12. Iterative examples
	12.1. Iterative examples and their properties
	12.2. Residual algebraic independence
	Exercises
Chapter 13. Approximating discrete valuation rings by regular local rings
	13.1. Local quadratic transforms and local uniformization
	13.2. Expressing a DVR as a directed union of regular local rings
	13.3. Approximating other rank-one valuation rings
	Exercises
Chapter 14. Non-Noetherian examples of dimension 3
	14.1. A family of examples in dimension 3
	14.2. Two cases of Examples 14.1 and their spectra
	Exercises
Chapter 15. Noetherian properties of non-Noetherian rings
	15.1. A general question and the setting for this chapter
	15.2. More properties of Insider Constructions
	15.3. A four-dimensional non-Noetherian domain
	Exercises
Chapter 16. Non-Noetherian examples in higher dimension
	16.1. Higher dimensional non-Noetherian examples
	16.2. A 4-dimensional prime spectrum
	16.3.