An algebraic introduction to K-theory:
This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and...
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2002
|
| Series: | Encyclopedia of mathematics and its applications
volume 87 |
| Subjects: | |
| Online Access: | DE-12 DE-92 DE-706 Volltext |
| Summary: | This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs |
| Item Description: | Title from publisher's bibliographic system (viewed on 05 Oct 2015) |
| Physical Description: | 1 online resource (xiv, 676 pages) |
| ISBN: | 9781107326002 |
| DOI: | 10.1017/CBO9781107326002 |
Staff View
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| 520 | |a This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs | ||
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| contents | Groups of Modules: K[subscript 0] Free Modules Bases Matrix Representations Absence of Dimension Projective Modules Direct Summands Summands of Free Modules Grothendieck Groups Semigroups of Isomorphism Classes Semigroups to Groups Resolutions Stability for Projective Modules Adding Copies of R Stably Free Modules When Stably Free Modules Are Free Stable Rank Dimensions of a Ring Multiplying Modules Semirings Burnside Rings Tensor Products of Modules Change of Rings K[subscript 0] of Related Rings G[subscript 0] of Related Rings K[subscript 0] as a Functor The Jacobson Radical Localization Sources of K[subscript 0] Number Theory Algebraic Integers Dedekind Domains Ideal Class Groups Extensions and Norms K[subscript 0] and G[subscript 0] of Dedekind Domains Group Representation Theory Linear Representations Representing Finite Groups Over Fields Semisimple Rings Characters Groups of Matrices: K[subscript 1] Definition of K[subscript 1] Elementary Matrices Commutators and K[subscript 1](R) Determinants The Bass K[subscript 1] of a Category Stability for K[subscript 1](R) Surjective Stability Injective Stability Relative K[subscript 1] Congruence Subgroups of GL[subscript n](R) Congruence Subgroups of SL[subscript n](R) Mennicke Symbols Relations Among Matrices: K[subscript 2] K[subscript 2](R) and Steinberg Symbols Definition and Properties of K[subscript 2](R) Elements of St(R) and K[subscript 2](R) Exact Sequences The Relative Sequence Excision and the Mayer-Vietoris Sequence The Localization Sequence Universal Algebras Presentation of Algebras Graded Rings The Tensor Algebra Symmetric and Exterior Algebras The Milnor Ring Tame Symbols Norms on Milnor K-Theory Matsumoto's Theorem Sources of K[subscript 2] Symbols in Arithmetic Hilbert Symbols Metric Completion of Fields The p-Adic Numbers and Quadratic Reciprocity Local Fields and Norm Residue Symbols Brauer Groups The Brauer Group of a Field Splitting Fields Twisted Group Rings The K[subscript 2] Connection A Sets, Classes, Functions Chain Conditions, Composition Series |
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| dewey-full | 512/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.55 |
| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1017/CBO9781107326002 |
| format | Electronic eBook |
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| id | DE-604.BV043941852 |
| illustrated | Not Illustrated |
| indexdate | 2024-10-02T10:01:36Z |
| institution | BVB |
| isbn | 9781107326002 |
| language | English |
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| series2 | Encyclopedia of mathematics and its applications |
| spelling | Magurn, Bruce A. Verfasser aut An algebraic introduction to K-theory Bruce A. Magurn Cambridge Cambridge University Press 2002 1 online resource (xiv, 676 pages) txt rdacontent c rdamedia cr rdacarrier Encyclopedia of mathematics and its applications volume 87 Title from publisher's bibliographic system (viewed on 05 Oct 2015) This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs K-theory K-Theorie (DE-588)4033335-8 gnd rswk-swf K-Theorie (DE-588)4033335-8 s 1\p DE-604 Erscheint auch als Druckausgabe 978-0-521-10658-0 Erscheint auch als Druckausgabe 978-0-521-80078-5 https://doi.org/10.1017/CBO9781107326002 Verlag URL des Erstveröffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Magurn, Bruce A. An algebraic introduction to K-theory Groups of Modules: K[subscript 0] Free Modules Bases Matrix Representations Absence of Dimension Projective Modules Direct Summands Summands of Free Modules Grothendieck Groups Semigroups of Isomorphism Classes Semigroups to Groups Resolutions Stability for Projective Modules Adding Copies of R Stably Free Modules When Stably Free Modules Are Free Stable Rank Dimensions of a Ring Multiplying Modules Semirings Burnside Rings Tensor Products of Modules Change of Rings K[subscript 0] of Related Rings G[subscript 0] of Related Rings K[subscript 0] as a Functor The Jacobson Radical Localization Sources of K[subscript 0] Number Theory Algebraic Integers Dedekind Domains Ideal Class Groups Extensions and Norms K[subscript 0] and G[subscript 0] of Dedekind Domains Group Representation Theory Linear Representations Representing Finite Groups Over Fields Semisimple Rings Characters Groups of Matrices: K[subscript 1] Definition of K[subscript 1] Elementary Matrices Commutators and K[subscript 1](R) Determinants The Bass K[subscript 1] of a Category Stability for K[subscript 1](R) Surjective Stability Injective Stability Relative K[subscript 1] Congruence Subgroups of GL[subscript n](R) Congruence Subgroups of SL[subscript n](R) Mennicke Symbols Relations Among Matrices: K[subscript 2] K[subscript 2](R) and Steinberg Symbols Definition and Properties of K[subscript 2](R) Elements of St(R) and K[subscript 2](R) Exact Sequences The Relative Sequence Excision and the Mayer-Vietoris Sequence The Localization Sequence Universal Algebras Presentation of Algebras Graded Rings The Tensor Algebra Symmetric and Exterior Algebras The Milnor Ring Tame Symbols Norms on Milnor K-Theory Matsumoto's Theorem Sources of K[subscript 2] Symbols in Arithmetic Hilbert Symbols Metric Completion of Fields The p-Adic Numbers and Quadratic Reciprocity Local Fields and Norm Residue Symbols Brauer Groups The Brauer Group of a Field Splitting Fields Twisted Group Rings The K[subscript 2] Connection A Sets, Classes, Functions Chain Conditions, Composition Series K-theory K-Theorie (DE-588)4033335-8 gnd |
| subject_GND | (DE-588)4033335-8 |
| title | An algebraic introduction to K-theory |
| title_alt | Groups of Modules: K[subscript 0] Free Modules Bases Matrix Representations Absence of Dimension Projective Modules Direct Summands Summands of Free Modules Grothendieck Groups Semigroups of Isomorphism Classes Semigroups to Groups Resolutions Stability for Projective Modules Adding Copies of R Stably Free Modules When Stably Free Modules Are Free Stable Rank Dimensions of a Ring Multiplying Modules Semirings Burnside Rings Tensor Products of Modules Change of Rings K[subscript 0] of Related Rings G[subscript 0] of Related Rings K[subscript 0] as a Functor The Jacobson Radical Localization Sources of K[subscript 0] Number Theory Algebraic Integers Dedekind Domains Ideal Class Groups Extensions and Norms K[subscript 0] and G[subscript 0] of Dedekind Domains Group Representation Theory Linear Representations Representing Finite Groups Over Fields Semisimple Rings Characters Groups of Matrices: K[subscript 1] Definition of K[subscript 1] Elementary Matrices Commutators and K[subscript 1](R) Determinants The Bass K[subscript 1] of a Category Stability for K[subscript 1](R) Surjective Stability Injective Stability Relative K[subscript 1] Congruence Subgroups of GL[subscript n](R) Congruence Subgroups of SL[subscript n](R) Mennicke Symbols Relations Among Matrices: K[subscript 2] K[subscript 2](R) and Steinberg Symbols Definition and Properties of K[subscript 2](R) Elements of St(R) and K[subscript 2](R) Exact Sequences The Relative Sequence Excision and the Mayer-Vietoris Sequence The Localization Sequence Universal Algebras Presentation of Algebras Graded Rings The Tensor Algebra Symmetric and Exterior Algebras The Milnor Ring Tame Symbols Norms on Milnor K-Theory Matsumoto's Theorem Sources of K[subscript 2] Symbols in Arithmetic Hilbert Symbols Metric Completion of Fields The p-Adic Numbers and Quadratic Reciprocity Local Fields and Norm Residue Symbols Brauer Groups The Brauer Group of a Field Splitting Fields Twisted Group Rings The K[subscript 2] Connection A Sets, Classes, Functions Chain Conditions, Composition Series |
| title_auth | An algebraic introduction to K-theory |
| title_exact_search | An algebraic introduction to K-theory |
| title_full | An algebraic introduction to K-theory Bruce A. Magurn |
| title_fullStr | An algebraic introduction to K-theory Bruce A. Magurn |
| title_full_unstemmed | An algebraic introduction to K-theory Bruce A. Magurn |
| title_short | An algebraic introduction to K-theory |
| title_sort | an algebraic introduction to k theory |
| topic | K-theory K-Theorie (DE-588)4033335-8 gnd |
| topic_facet | K-theory K-Theorie |
| url | https://doi.org/10.1017/CBO9781107326002 |
| work_keys_str_mv | AT magurnbrucea analgebraicintroductiontoktheory |