Partially ordered rings and semi-algebraic geometry / Gregory W. Brumfiel.

Format:

Book

Author/Creator:

Brumfiel, Gregory W., author.

Series:

London Mathematical Society lecture note series ; 37.

London Mathematical Society lecture note series ; 37

Language:

English

Subjects (All):

Commutative rings.

Categories (Mathematics).

Physical Description:

1 online resource (280 pages) : digital, PDF file(s).

Other Title:

Partially Ordered Rings & Semi-Algebraic Geometry

Place of Publication:

Cambridge : Cambridge University Press, 1979.

Language Note:

English

Summary:

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.

Contents:

Cover; Title; Copyright; Contents; INTRODUCTION; CHAPTER I - PARTIALLY ORDERED RINGS; 1.1. Definitions; 1.2. Existence of Orders; 1.3. Extension and Contraction of Orders; 1.4. Simple refinements of orders; 1.5. Remarks on the Categories (PORNN) and (PORCK); 1.6. Remarks on Integral Domains; 1.7. Some Examples; CHAPTER II - HOMOMORPIIISMS AND CONVEX IDEALS; 2.1. Convex Ideals and Quotient Rings; 2.2. Convex Hulls; 2.3. Maximal Convex Ideals and Prime Convex Ideals; 2.4. Relation between Convex Ideals in (A,β) and (A/I, β/I); 2.5. Absolutely Convex Ideals; 2.6. Semi-Noetherian Rings

2.7. Convex Ideals and Intersections of Orders2.8. Some Examples; CHAPTER III- LOCALIZATION; 3.1. Partial Orders on Localized Rings; 3.2. Sufficiency of Positive Multiplicative Sets; 3.3. Refinements of an Order Induced by Certain Localizations; 3.4. Convex Ideals in (A,β) and (AT,βT); 3.5. Concave Multiplicative Sets; 3.6. The Shadow of 1; 3.7. Localization at a Prime Convex Ideal; 3.8. Localization in (PORCK); 3.9. Applications of Localization, I - Some Properties of Convex Prime Ideals; 3.10. Applications of Localization, II - Zero Divisors

3.11. Applications of Localization, III - Minimal Primes, Isolated Sets of Primes, and Associated Invariants3.12. Operators on the Set of Orders on a Ring; CHAPTER IV - SOME CATEGORICAL NOTIONS; 4.1. Fibre Products; 4.2. Fibre Sums; 4.3. Direct and Inverse Limits; 4.4. Some Examples; CHAPTER V - THE PRIME CONVEX IDEAL SPECTRUM; 5.1. The Zariski Topology Defined; 5.2. Some Topological Properties; 5.3. Irreducible Closed Sets in Spec(A,β); 5.4. Spec(A,β) as a Functor; 5.5. Disconnectedness of Spec(A,β); 5.6. The Structure Sheaf, I - A First Approximation on Basic Open Sets

5.7. The Structure Sheaf, II - The Sheaf Axioms for Basic Open Sets5.8. The Structure Sheaf, III - Definition; CHAPTER VI - POLYNOMIALS; 6.1. Polynomials as Functions; 6.2. Adjoining Roots; 6.3. A Universal Bound on the Roots of Polynomials; 6.4. A ""Going-Up"" Theorem for Semi-Integral Extensions; CHAPTER VII - ORDERED FIELDS; 7.1. Basic Results; 7.2. Function Theoretic Properties of Polynomials; 7.3. Sturm's Theorem; 7.4. Dedekind Cuts; Archimedean and Non-Archimedean Extensions.; 7.5. Orders on Simple Field Extensions; 7.6. Total Orders and Signed Places; 7.7. Existence of Signed Places

CHAPTER VIII - AFFINE SEMT-ALGEBRAIC SETS8.1. Introduction and Notation; 8.2. Some Properties of RHJ-Algebras; 8.3. Real Curves; 8.4. Signed Places on Function Fields; 8.5. Characterization of Non-Negative Functions; 8.6. Derived Orders; 8.7. A Preliminary Inverse Function Theorem; 8.8. Algebraic Simple Points, Dimension, Codimension and Rank; 8.9. Stratification of Semi-Algebraic Sets; 8.10. Krull Dimension; 8.11. Orders on Function Fields; 8.12. Discussion of Total Orders on R(x,y); 8.13. Brief Discussion of Structure Sheaves; I - The rational structure sheaf

II - The semi-algebraic structure sheaf

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-88380-1

1-107-36583-X

1-107-37056-6

1-107-36092-7

1-107-36994-0

1-299-40364-6

1-107-36337-3

0-511-89192-X

0-511-72153-6