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Introduction to Algebraic K-Theory
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John Milnor
Language:
English
Published/Copyright:
2016
About this book
Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.
Reviews
"John Willard Milnor, Winner of the 2011 Leroy P. Steele Prize for Lifetime Achievement, American Mathematical Society"
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"John Milnor, Winner of the 2011 Abel Prize from the Norwegian Academy of Science and Letters"
Topics
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Frontmatter
i -
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Preface and Guide to the Literature
vii -
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Contents
xiii -
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§1. Projective Modules and K0Λ
1 -
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§2 . Constructing Projective Modules
19 -
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§3. The Whitehead Group K1Λ
25 -
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§4. The Exact Sequence Associated with an Ideal
33 -
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§5. Steinberg Groups and the Functor K2
39 -
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§6. Extending the Exact Sequences
53 -
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§7. The Case of a Commutative Banach Algebra
57 -
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§8. The Product K1Λ ⊗ K1Λ → K2Λ
63 -
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§9. Computations in the Steinberg Group
71 -
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§10. Computation of K2Z
81 -
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§11. Matsumoto’s Computation of K2 of a Field
93 -
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12. Proof of Matsumoto’s Theorem
109 -
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§13. More about Dedekind Domains
123 -
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§14. The Transfer Homomorphism
137 -
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§15. Power Norm Residue Symbols
143 -
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§16. Number Fields
155 -
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Appendix. Continuous Steinberg Symbols
165 -
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Index
183
Publishing information
Pages and Images/Illustrations in book
eBook published on:
June 20, 2016
eBook ISBN:
9781400881796
Pages and Images/Illustrations in book
eBook ISBN:
9781400881796
Keywords for this book
Homomorphism; Root of unity; Elementary matrix; Subgroup; Dedekind domain; K-theory; Topological group; Complex number; Ideal (ring theory); Exterior algebra; Algebraic K-theory; Banach algebra; Vector space; Wedderburn's theorem; Ring of integers; Algebraic integer; Rational number; Division algebra; Basis (linear algebra); Cyclic group; Local field; Commutative property; Commutative ring; Direct sum; Monomial; Function (mathematics); Theorem; Quotient ring; Real number; Topological K-theory; Existential quantification; Special linear group; Variable (mathematics); Number theory; Isomorphism class; Matsumoto's theorem; Integral domain; Galois extension; Maximal ideal; Algebraic equation; General linear group; Surjective function; Homological algebra; Identity matrix; Congruence subgroup; Topological space; Coprime integers; Noetherian; Polynomial; Fundamental group; Prime ideal; Tensor product; Projective module; Division ring; Invertible matrix; Exact sequence; Kummer theory; Simple algebra; Commutator; Identity element; Abelian group; Absolute value; Topology; Special case; Hausdorff space; Prime element; Free abelian group; Direct limit; Scientific notation; Mathematics
Audience(s) for this book
For an expert adult audience, including professional development and academic research
