Special groups : boolean-theoretic methods in the theory of quadratic forms / M.A. Dickmann, F. Miraglia.

Format:

Book

Author/Creator:

Dickmann, M. A., 1940- author.

Miraglia, Francisco, author.

Series:

Memoirs of the American Mathematical Society ; no. 689.

Memoirs of the American Mathematical Society, 0065-9266 ; number 689

Language:

English

Subjects (All):

Forms, Quadratic.

Algebra, Boolean.

Physical Description:

1 online resource (271 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2000.

Language Note:

English

Summary:

This monograph presents a systematic study of Special Groups, a first-order universal-existential axiomatization of the theory of quadratic forms, which comprises the usual theory over fields of characteristic different from 2, and is dual to the theory of abstract order spaces. The heart of our theory begins in Chapter 4 with the result that Boolean algebras have a natural structure of reduced special group. More deeply, every such group is canonically and functorially embedded in a certain Boolean algebra, its Boolean hull. This hull contains a wealth of information about the structure of the given special group, and much of the later work consists in unveiling it. Thus, in Chapter 7 we introduce two series of invariants "living" in the Boolean hull, which characterize the isometry of forms in any reduced special group. While the multiplicative series--expressed in terms of meet and symmetric difference--constitutes a Boolean version of the Stiefel-Whitney invariants, the additive series--expressed in terms of meet and join--, which we call Horn-Tarski invariants, does not have a known analog in the field case; however, the latter have a considerably more regular behaviour. We give explicit formulas connecting both series, and compute explicitly the invariants for Pfister forms and their linear combinations. In Chapter 9 we combine Boolean-theoretic methods with techniques from Galois cohomology and a result of Voevodsky to obtain an affirmative solution to a long standing conjecture of Marshall concerning quadratic forms over formally real Pythagorean fields. Boolean methods are put to work in Chapter 10 to obtain information about categories of special groups, reduced or not. And again in Chapter 11 to initiate the model-theoretic study of the first-order theory of reduced special groups, where, amongst other things we determine its model-companion. The first-order approach is also present in the study of some outstanding classes of morphisms carried out in Chapter 5, e.g., the pure embeddings of special groups. Chapter 6 is devoted to the study of special groups of continuous functions.

Contents:

""Contents""; ""Preface""; ""Chapter 1. Special Groups""; ""1. Special Groups and their Morphisms. Examples""; ""2. Characterizations of Special Groups""; ""3. Fields and Special Groups""; ""Chapter 2. Pfister Forms, Saturated Subgroups and Quotients""; ""1. Pfister Forms""; ""2. Saturated Subgroups""; ""3. Pfister quotients""; ""Chapter 3. The Space of Orders of a Reduced Group. Duality""; ""Chapter 4. Boolean Algebras and Reduced Special Groups""; ""1. Boolean Algebras as Reduced Special Groups""; ""2. The Boolean Hull of a Reduced Special Group""; ""Chapter 5. Embeddings""

""1. Complete Embeddings""""2. SG-morphisms with a retract""; ""3. Pure Embeddings""; ""4. Isotropy-Reflecting Morphisms""; ""Chapter 6. Special Groups of Continuous Functions""; ""1. Filtered Powers""; ""2. SG-filtered Powers""; ""3. Stalks and Germs""; ""4. The Space of Orders of a SG-filtered Power""; ""5. The Boolean Hull of SG-Filtered Powers""; ""Chapter 7. Horn-Tarski and Stiefel-Whitney Invariants""; ""1. The Horn-Tarski conditions""; ""2. Basic Properties""; ""3. Some applications""; ""4. The invariants of Pfister forms""; ""5. The invariants of linear combinations of Pfister forms""

""Chapter 8. Algebraic K-theory of Fields and Special Groups""""1. Milnor's Algebraic K-theory""; ""2. Isometry and K-Theoretic Stiefel-Whitney Invariants""; ""Chapter 9. Marshall's Conjecture for Pythagorean Fields""; ""1. Introduction""; ""2. The inductive limit of graded rings of exponent two""; ""3. The isomorphisms k(F) â?? B(F) and W(G) â?? B[sub(G)]""; ""4. The equivalence of Marshall's conjecture to the weak Marshall conjecture for AV groups""; ""5. Marshall's conjecture for Pythagorean fields""; ""Chapter 10. The category of special groups""; ""1. Monics and Epics""

""2. Free, injective and projective objects""""3. Products and Coproducts""; ""Chapter 11. Some Model Theory of Special Groups""; ""1. The first-order theory of fans""; ""2. Existentially closed special groups""; ""3. The model-completion of the theory of reduced special groups""; ""4. The atomless hull of a Boolean algebra""; ""5. The atomless hull of a reduced special group""; ""6. Quantifier Elimination""; ""Appendix A. The Universal Theory of Reduced Special Groups""; ""Appendix B. Table of References for [DM1] and [DM2]""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""

""F""""G""; ""H""; ""I""; ""K""; ""L""; ""M""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""

Notes:

"May 2000, volume 145, number 689 (second of 4 numbers)."

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-0280-7