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Integral Domains Inside Noetherian Power Series Rings: Constructions and Examples May 31, 2020

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Integral Domains Inside Noetherian Power Series Rings: Constructions and Examples May 31, 2020 William Heinzer Christel Rotthaus Sylvia Wiegand Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 E-mail address: [email protected] Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027 E-mail address: [email protected] Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130 E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary 13A15, 13B35, 13F25; Secondary 13B02, 13B22, 13B40, 13C15, 13H05 Key words and phrases. power series, Noetherian integral domain, completion, generic fiber, flatness, prime spectra. Abstract. A major theme of this monograph is the creation of examples that are appropriate intersections of a field with a homomorphic image of a power series ring over a Noetherian domain. Classical examples of Noetherian integral domains with interesting properties are constructed by Akizuki, Schmidt, and Nagata. This work is continued by Brodmann-Rotthaus, Ferrand-Raynaud, Heitmann, Lequain, McAdam, Nishimura, Ogoma, Ratliff, Rotthaus, Weston and others. In certain circumstances, the intersection examples may be realized as a directed union, and the Noetherian property for the associated directed union is equivalent to a flatness condition. This flatness criterion simplifies the anal- ysis of several classical examples and yields other examples such as • A catenary Noetherian local integral domain of any specified dimension of at least two that has geometrically regular formal fibers and is not universally catenary. • A three-dimensional non-Noetherian unique factorization domain B such that the unique maximal ideal of B has two generators; B has precisely n prime ideals of height two, where n is an arbitrary positive integer; and each prime ideal of B of height two is not finitely generated but all the other prime ideals of B are finitely generated. • A two-dimensional Noetherian local domain that is a birational extension of a polynomial ring in three variables over a field yet fails to have Cohen- Macaulay formal fibers. This example also demonstrates that Serre's condition S1 need not lift to the completion; the example is related to an example of Ogoma. Another theme is an analysis of extensions of integral domains R,! S having trivial generic fiber, that is, every nonzero prime ideal of S has a nonzero intersection with R. Motivated by a question of Hochster and Yao, we present results about • The height of prime ideals maximal in the generic fiber of certain exten- sions involving mixed polynomial-power series rings. • The prime ideal spectrum of a power series ring in one variable over a one-dimensional Noetherian domain. • The dimension of S if R,! S is a local map of complete local domains having trivial generic fiber. A third theme relates to the questions: • What properties of a Noetherian domain extend to a completion? • What properties of an ideal pass to its extension in a completion? • What properties extend for a more general multi-adic completion? We give an example of a three-dimensional regular local domain R having a prime ideal P of height two with the property that the extension of P to the completion of R is not integrally closed. All of these themes are relevant to the study of prime spectra of Noether- ian rings and of the induced spectral maps associated with various extensions of Noetherian rings. We describe the prime spectra of various extensions in- volving power series. Contents Preface ix Chapter 1. Introduction 1 Chapter 2. Tools 9 2.1. Conventions and terminology 9 2.2. Basic theorems 16 2.3. Flatness 18 Exercises 21 Chapter 3. More tools 25 3.1. Introduction to ideal-adic completions 25 3.2. Uncountable transcendence degree for a completion 30 3.3. Power series pitfalls 31 3.4. Basic results about completions 32 3.5. Chains of prime ideals, fibers of maps 34 3.6. Henselian rings 36 3.7. Regularity and excellence 38 Exercises 41 Chapter 4. First examples of the construction 45 4.1. Universality 45 4.2. Elementary examples 47 4.3. Historical examples 48 4.4. Prototypes 51 Exercises 55 Chapter 5. The Inclusion Construction 59 5.1. The Inclusion Construction and a picture 59 5.2. Approximations for the Inclusion Construction 60 5.3. Basic properties of the constructed domains 63 Exercises 71 Chapter 6. Flatness and the Noetherian property 73 6.1. The Noetherian Flatness Theorem 73 6.2. Introduction to Insider Construction 10.7 81 6.3. Nagata's example 82 6.4. Christel's Example 83 6.5. Further implications of the Noetherian Flatness Theorem 84 Exercise 85 v vi CONTENTS Chapter 7. The flat locus of an extension of polynomial rings 87 7.1. Flatness criteria 87 7.2. The Jacobian ideal and the smooth and flat loci 91 7.3. Applications to polynomial extensions 98 Exercises 101 Chapter 8. Excellent rings and formal fibers 103 8.1. Background for excellent rings 103 8.2. Nagata rings and excellence 109 8.3. Henselian rings 111 Exercises 113 Chapter 9. Height-one prime ideals and weak flatness 115 9.1. The limit-intersecting condition 115 9.2. Height-one primes in extensions of Krull domains 119 9.3. Flatness for Inclusion Construction 5.3 122 Exercises 124 Chapter 10. Insider Construction details 127 10.1. Localized polynomial rings over special DVRs 128 10.2. Describing the construction 130 10.3. The non-flat locus of Insider Construction 10.7 131 10.4. Preserving excellence with the Insider Construction 136 Chapter 11. Integral closure under extension to the completion (intclsec) 139 11.1. Integral closure under ring extension 139 11.2. Extending ideals to the completion 142 11.3. Comments and Questions 145 Exercises 147 Chapter 12. Iterative examples 149 12.1. Iterative examples and their properties 149 12.2. Residual algebraic independence 154 Exercises 162 Chapter 13. Approximating discrete valuation rings by regular local rings 165 13.1. Local quadratic transforms and local uniformization 165 13.2. Expressing a DVR as a directed union of regular local rings 167 13.3. Approximating other rank-one valuation rings 171 Exercises 171 Chapter 14. Non-Noetherian examples of dimension 3 173 14.1. A family of examples in dimension 3 173 14.2. Two cases of Examples 14.1 and their spectra 177 Exercises 183 Chapter 15. Noetherian properties of non-Noetherian rings 185 15.1. A general question and the setting for this chapter 185 15.2. More properties of Insider Constructions 187 15.3. A four-dimensional non-Noetherian domain 189 Exercises 196 CONTENTS vii Chapter 16. Non-Noetherian examples in higher dimension 197 16.1. Higher dimensional non-Noetherian examples 197 16.2. A 4-dimensional prime spectrum 200 16.3. D + M constructions, coherence, finite conductor domains 204 Exercises 207 Chapter 17. The Homomorphic Image Construction 209 17.1. Two construction methods and a picture 209 17.2. Approximations for Homomorphic Image Construction 211 17.3. Basic properties of the Approximation Domains 213 17.4. Noetherian flatness for homomorphic images 215 17.5. Weak Flatness for Homomorphic Image Construction 17.2 218 17.6. Inclusion Constructions are Homomorphic Image Constructions 219 17.7. Connecting the two constructions 224 Exercises 229 Chapter 18. Catenary local rings with geometrically normal formal fibers 231 18.1. History, terminology and summary 231 18.2. Geometrically normal formal fibers and the catenary property 233 18.3. Flatness for the intersection of finitely many ideals 237 18.4. Regular maps and geometrically regular formal fibers 238 18.5. Examples that are not universally catenary 240 18.6. The depth of the constructed rings 244 Exercises 244 Chapter 19. An Ogoma-like example 245 19.1. Cohen-Macaulay formal fibers 245 19.2. Constructing the Ogoma-like example 252 Chapter 20. Multi-ideal-adic completions of Noetherian rings 257 20.1. Ideal filtrations and completions 257 20.2. Basic mechanics for the multi-adic completion 258 20.3. Preserving Noetherian under multi-adic completion 262 20.4. Preserving excellence or Henselian under multi-adic completion 265 Exercises 267 Chapter 21. Noetherian flatness and multi{adic constructions 269 21.1. Flatness lemmas 269 21.2. Multi-adic inclusion constructions 270 Chapter 22. Idealwise algebraic independence, 275 22.1. Idealwise independence, weakly flat and PDE extensions 275 22.2. Primarily independent elements 280 22.3. Residually algebraically independent elements 285 Exercises 293 Chapter 23. Idealwise algebraic independence II 295 23.1. Primary independence and flatness 295 23.2. Composition, base change and polynomial extensions 297 23.3. Passing to the Henselization 302 23.4. Summary diagram for the independence concepts 306 viii CONTENTS Chapter 24. Krull domains with Noetherian x-adic completions 309 24.1. Applying Inclusion Construction 5.3 310 24.2. Limit-intersecting elements 311 24.3. Example where the intersection domain equals the approximation domain and is non-Noetherian 317 24.4. A birational connection and additional examples 324 Exercises 327 Chapter 25. Inclusion Constructions over excellent normal local domains 329 25.1. Primarily limit-intersecting extensions and flatness 329 25.2. Existence of primarily limit-intersecting extensions 332 Exercise 339 Chapter 26. Weierstrass techniques for generic fiber rings 341 26.1. Terminology, background and results 341 26.2. Variations on a theme by Weierstrass 345 Exercises 355 Chapter 27. Generic fiber rings of mixed polynomial-power series rings 357 27.1. Weierstrass implications for the ring B = k[[X]] [Y ](X;Y ) 357 27.2. Weierstrass implications for the ring C = k[Y ](Y )[[X]] 360 27.3. Subrings of the power series ring k[[z; t]] 363 27.4. Weierstrass implications for the localized polynomial ring A 366 27.5.
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