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. |
Contents
R Shafarevich | 5 |
2 Fields | 22 |
7 The Algebraic View of Infinitesimal Notions | 50 |
Copyright | |
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Common terms and phrases
a₁ Abelian group algebras of finite analogue arbitrary automorphism axioms b₁ C₁ called central division algebras coefficients cohomology commutative ring compact complex analytic complex number consider consists construction contained coordinate coordinatisation corresponding coset decomposition defined definition denoted dimension direct sum division algebra e₁ elements equation Example exists finite group finite length finite number finite rank finite-dimensional follows functor G₁ G₂ Galois geometry given GL(n group G Hence homomorphism identity integers invariant inverse irreducible representations isomorphic kernel lattice Lie algebra Lie groups linear transformations manifold matrix module morphisms multiplication n-dimensional normal subgroup notion number field Obviously orthogonal transformations p₁ permutations plane polynomial properties quaternions quotient rational functions real number realised relations Riemann surface satisfying semisimple sequence simple solvable submodule subspace symmetry group tangent tensor Theorem theory tions topological space valuation vector fields vector space vertexes