## Basic Notions of Algebra§22. K-theory 230 A. Topological X-theory 230 Vector bundles and the functor Vec(X). Periodicity and the functors KJX). K(X) and t the infinite-dimensional linear group. The symbol of an elliptic differential operator. The index theorem. B. Algebraic K-theory 234 The group of classes of projective modules. K , K and K of a ring. K of a field and o l n 2 its relations with the Brauer group. K-theory and arithmetic. Comments on the Literature 239 References 244 Index of Names 249 Subject Index 251 Preface This book aims to present a general survey of algebra, of its basic notions and main branches. Now what language should we choose for this? In reply to the question 'What does mathematics study?', it is hardly acceptable to answer 'structures' or 'sets with specified relations'; for among the myriad conceivable structures or sets with specified relations, only a very small discrete subset is of real interest to mathematicians, and the whole point of the question is to understand the special value of this infinitesimal fraction dotted among the amorphous masses. In the same way, the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion. |

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### Contents

1 What is Algebra? | 6 |

2 Fields | 11 |

3 Commutative Rings | 17 |

4 Homomorphisms and Ideals | 24 |

5 Modules | 33 |

6 Algebraic Aspects of Dimension | 41 |

7 The Algebraic View of Infinitesimal Notions | 50 |

8 Noncommutative Rings | 61 |

C Representations of the Classical Complex Lie Groups | 176 |

18 Some Applications of Groups | 178 |

B The Galois Theory of Linear Differential Equations PicardVessiot Theory | 182 |

C Classification of Unramified Covers | 183 |

D Invariant Theory | 184 |

E Group Representations and the Classification of Elementary Particles | 186 |

19 Lie Algebras and Nonassociative Algebra | 189 |

B Lie Theory | 193 |

9 Modules over Noncommutative Rings | 74 |

10 Semisimple Modules and Rings | 79 |

11 Division Algebras of Finite Rank | 90 |

12 The Notion of a Group | 96 |

Finite Groups | 108 |

Infinite Discrete Groups | 125 |

Lie Groups and Algebraic Groups | 141 |

A Compact Lie Groups | 144 |

B Complex Analytic Lie Groups | 148 |

C Algebraic Groups | 151 |

16 General Results of Group Theory | 152 |

17 Group Representations | 161 |

A Representations of Finite Groups | 164 |

B Representations of Compact Lie Groups | 168 |

C Applications of Lie Algebras | 198 |

D Other Nonassociative Algebras | 200 |

20 Categories | 203 |

21 Homological Algebra | 214 |

B Cohomology of Modules and Groups | 220 |

C Sheaf Cohomology | 226 |

22 Ktheory | 231 |

B Algebraic Ktheory | 235 |

Comments on the Literature | 240 |

References | 245 |

250 | |

252 | |

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Abelian groups algebraic groups algebras of finite analogue arbitrary automorphism axioms called coefficients cohomology commutative ring compact complex analytic complex numbers consider consists construction contained coordinate coordinatisation corresponding coset defined definition denoted differential operators dimension dimensional direct sum division algebra elements equation exact sequence Example exists extension L/K finite extension finite fields finite groups finite length finite number finite rank finite type finite-dimensional follows functor g e G Galois geometry given GL(n group G Hence homomorphism identity integral invariant irreducible representations isomorphic kernel lattice Lie algebra Lie groups linear transformations manifold matrix morphisms multiplication n-dimensional Noetherian normal subgroup notion number field permutations plane polynomial ring properties quaternions quotient quotient group rational functions real number relations satisfying semisimple sheaf simple SL(n solvable submodule subspace symmetry group tensor Theorem theory topological space unique vector fields vector space