The purpose of this unique book is to establish purely algebraic foundations for the development of certain parts of topology. Some topologists seek to understand geometric properties of solutions to finite systems of equations or inequalities and configurations which in some sense actually occur in the real world. Others study spaces constructed more abstractly using infinite limit processes. Their goal is to determine just how similar or different these abstract spaces are from those which are finitely described. However, as topology is usually taught, even the first, more concrete type of problem is approached using the language and methods of the second type. Professor Brumfiel's thesis is that this is unnecessary and, in fact, misleading philosophically. He develops a type of algebra, partially ordered rings, in which it makes sense to talk about solutions of equations and inequalities and to compare geometrically the resulting spaces. The importance of this approach is primarily that it clarifies the sort of geometrical questions one wants to ask and answer about those spaces which might have physical significance.
Author(s): Gregory W. Brumfiel
Series: London Mathematical Society Lecture Note Series 37
Publisher: Cambridge University Press
Year: 1980
Language: English
Pages: 292
Cover......Page 1
London Mathematical Society Lecture Note Series 37......Page 2
Partially Ordered Rings and Semi-Algebraic Geometry......Page 4
9780521228459......Page 5
Contents......Page 6
Preface......Page 10
Introduction......Page 12
1.1. Definitions......Page 43
1.2. Existence of Orders......Page 44
1.3. Extension and Contraction of Orders......Page 45
1.4. Simple refinements of orders......Page 47
1.5. Remarks on the Categories (PORNN) and (PORCK)......Page 48
1.6. Remarks on Integral Domains......Page 50
1.7. Some Examples......Page 51
2.1. Convex Ideals and Quotient Rings......Page 56
2.2. Convex Hulls......Page 57
2.3. Maximal Convex Ideals and Prime Convex Ideals......Page 60
2.4. Relation between Convex Ideals in (A,β) and (A/I, β/I)......Page 63
2.6. Semi-Noetherian Rings......Page 67
2.7. Convex Ideals and Intersections of Orders......Page 73
2.8. Some Examples......Page 77
3.1. Partial Orders on Localized Rings......Page 88
3.2. Sufficiency of Positive Multiplicative Sets......Page 90
3.3. Refinements of an Order Induced by Certain Localizations......Page 91
3.4. Convex Ideals in (A, β) and (A_T, β_T)......Page 92
3.5. Concave Multiplicative Sets......Page 94
3.6. The Shadow of 1......Page 95
3.7. Localization at a Prime Convex Ideal......Page 98
3.8. Localization in (PORCK)......Page 99
3.9. Applications of Localization, I - Some Properties of Convex Prime Ideals......Page 100
3.10. Applications of Localization, II - Zero Divisors......Page 102
3.11. Applications of Localization, III - Minimal Primes, Isolated Sets of Primes, and Associated Invariants......Page 104
3.12. Operators on the Set of Orders on a Ring......Page 107
4.1. Fibre Products......Page 112
4.2. Fibre Sums......Page 113
4.3. Direct and Inverse Limits......Page 114
4.4. Some Examples......Page 115
5.1. The Zariski Topology Defined......Page 117
5.3. Irreducible Closed Sets in Spec(A,β)......Page 118
5.4. Spec(A,β) as a Functor......Page 120
5.6. The Structure Sheaf, I - A First Approximation on Basic Open Sets......Page 123
5.7. The Structure Sheaf, II - The Sheaf Axioms for Basic Open Sets......Page 124
5.8. The Structure Sheaf, ΙΙΙ - Definition......Page 126
6.1. Polynomials as Functions......Page 129
6.2. Adjoining Roots......Page 131
6.3. A Universal Bound on the Roots of Polynomials......Page 134
6.4. A "Going-Up" Theorem for Semi-Integral Extensions......Page 136
7.1. Basic Results......Page 141
7.2. Function Theoretic Properties of Polynomials......Page 143
7.3. Sturm's Theorem......Page 146
7.4. Dedekind Cuts; Archimedean and Non-Archimedean Extensions......Page 148
7.5. Orders on Simple Field Extensions......Page 151
7.6. Total Orders and Signed Places......Page 155
7.7. Existence of Signed Places......Page 159
8.1. Introduction and Notation......Page 173
8.2. Some Properties of RHJ-Algebras......Page 179
8.3. Real Curves......Page 189
8.4. Signed Places on Function Fields......Page 195
8.5. Characterization of Non-Negative Functions......Page 204
8.6. Derived Orders......Page 207
8.7. A Preliminary Inverse Function Theorem......Page 217
8.8. Algebraic Simple Points, Dimension, Codimension and Rank......Page 223
8.9. Stratification of Semi-Algebraic Sets......Page 229
8.10. Krull Dimension......Page 235
8.11. Orders on Function Fields......Page 243
8.12. Discussion of Total Orders on R(x,y)......Page 251
8.13. Brief Discussion of Structure Sheaves......Page 258
I - The rational structure sheaf......Page 259
II - The semi-algebraic structure sheaf......Page 263
IΙΙ - The smooth structure sheaf......Page 273
APPENDIX - The Tarski-Seidenberg Theorem......Page 279
BIBLIOGRAPHY......Page 284
LIST OF NOTATION......Page 289
INDEX......Page 290
