Valuations, orderings, and Milnor K-theory / Ido Efrat.

Format:

Book

Author/Creator:

Efrat, Ido, 1963- author.

Series:

Mathematical surveys and monographs ; no. 124.

Mathematical surveys and monographs, 0076-5376 ; volume 124

Language:

English

Subjects (All):

Valuation theory.

Ordered fields.

K-theory.

Physical Description:

1 online resource (305 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2006]

Language Note:

English

Summary:

This monograph is a comprehensive exposition of the modern theory of valued and ordered fields. It presents the classical aspects of such fields: their arithmetic, topology, and Galois theory. Deeper cohomological aspects are studied in its last part in an elementary manner. This is done by means of the newly developed theory of generalized Milnor $K$-rings. The book emphasizes the close connections and interplay between valuations and orderings, and to a large extent, studies themin a unified manner. The presentation is almost entirely self-contained. In particular, the text develops the needed machinery of ordered abelian groups. This is then used throughout the text to replace the more classical techniques of commutative algebra. Likewise, the book provides an introductionto the Milnor $K$-theory. The reader is introduced to the valuation-theoretic techniques as used in modern Galois theory, especially in applications to birational anabelian geometry, where one needs to detect valuations from their ``cohomological footprints''. These powerful techniques are presented here for the first time in a unified and elementary way.

Contents:

""Contents""; ""Introduction""; ""Conventions""; ""Part I. Abelian Groups""; ""Chapter 1. Preliminaries on Abelian Groups""; ""Â1.1. General facts""; ""Â1.2. Divisible hulls""; ""Â1.3. Rational ranks""; ""Â1.4. Characters""; ""Chapter 2. Ordered Abelian Groups""; ""Â2.1. Basic properties and examples""; ""Â2.2. Ranks""; ""Â2.3. Cores""; ""Â2.4. Cofinality and infinitesimals""; ""Â2.5. Ordered abelian groups of rank 1""; ""Â2.6. Push-downs""; ""Â2.7. Well-ordered sets""; ""Â2.8. Formal power series""; ""Â2.9. Generalized rational functions""; ""Part II. Valuations and Orderings""

""Chapter 3. Valuations""""Â3.1. Valuation rings""; ""Â3.2. Valuations""; ""Â3.3. Places""; ""Â3.4. Discrete valuations""; ""Chapter 4. Examples of Valuations""; ""Â4.1. Valuations from unique factorization domains""; ""Â4.2. Valuations on power series fields""; ""Â4.3. Gauss valuations""; ""Chapter 5. Coarsenings of Valuations""; ""Â5.1. Coarser and finer""; ""Â5.2. Quotients and compositions of valuations""; ""Â5.3. Coarsenings in the mixed characteristic case""; ""Chapter 6. Orderings""; ""Â6.1. Ordered fields""; ""Â6.2. Examples of orderings""; ""Â6.3. Archimedean orderings""

""Chapter 7. The Tree of Localities""""Â7.1. Localities""; ""Â7.2. Localities on residue fields""; ""Â7.3. The tree structure""; ""Chapter 8. Topologies""; ""Â8.1. Basic properties""; ""Â8.2. Continuity of roots""; ""Â8.3. Bounded sets""; ""Chapter 9. Complete Fields""; ""Â9.1. Metrics""; ""Â9.2. Examples""; ""Â9.3. Completions""; ""Chapter 10. Approximation Theorems""; ""Â10.1. Approximation by independent localities""; ""Â10.2. Approximation by incomparable valuations""; ""Â10.3. Consequences""; ""Chapter 11. Canonical Valuations""; ""Â11.1. Compatible localities""

""Â11.2. S-cores""""Â11.3. Explicit constructions""; ""Â11.4. Existence of valuations""; ""Chapter 12. Valuations of Mixed Characteristics""; ""Â12.1. Multiplicative representatives""; ""Â12.2. λ-adic expansions""; ""Â12.3. p-perfect structures""; ""Â12.4. Rings of Witt vectors""; ""Â12.5. Mixed valuations under a finiteness assumption""; ""Part III. Galois Theory""; ""Chapter 13. Infinite Galois Theory""; ""Chapter 14. Valuations in Field Extensions""; ""Â14.1. Chevalley's theorem""; ""Â14.2. Valuations in algebraic extensions""; ""Â14.3. The Galois action""

""Chapter 15. Decomposition Groups""""Â15.1. Definition and basic properties""; ""Â15.2. Immediateness of decomposition fields""; ""Â15.3. Relatively Henselian fields""; ""Chapter 16. Ramification Theory""; ""Â16.1. Inertia groups""; ""Â16.2. Ramification groups""; ""Chapter 17. The Fundamental Equality""; ""Â17.1. The fundamental inequality""; ""Â17.2. Ostrowski's theorem""; ""Â17.3. Defectless fields""; ""Â17.4. Extensions of discrete valuations""; ""Chapter 18. Hensel's Lemma""; ""Â18.1. The main variants""; ""Â18.2. nth powers""; ""Â18.3. Example: complete valued fields""

""Â18.4. Example: power series fields""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 269-274) and index.

Description based on print version record.

ISBN:

1-4704-1351-5