## Algebraic K-Theory and Its ApplicationsAlgebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory. |

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### Contents

I | 1 |

II | 7 |

III | 11 |

IV | 16 |

V | 27 |

VI | 32 |

VII | 41 |

VIII | 59 |

XXV | 240 |

XXVI | 245 |

XXVII | 265 |

XXVIII | 279 |

XXX | 281 |

XXXI | 289 |

XXXII | 295 |

XXXIII | 302 |

### Common terms and phrases

acyclic algebra assembly map basepoint BGL(R central extension chain complex Chern character cohomology commutative diagram commutative ring compact construction Corollary corresponding CW-complex cyclic group cyclic homology Dedekind domain Deduce defined Definition diagonal dimension direct sum direct summand double complex element elementary example Exercise fact fibration field finite type finitely generated projective free abelian group functor fundamental group gives GL(n GL(R hence Hochschild Hochschild homology homology groups homomorphism homotopy equivalence i?-module idempotent identity if-theory inclusion induces an isomorphism injective integers inverse kernel Ki(R Ko(R Kq(R left regular Lemma Let G matrix Mn(R morphism multiplication non-trivial non-zero Note operator polynomial prime ideals projective modules proof of Theorem Proposition quotient map R-modules relations short exact sequence SKi(R SL(n split Steinberg symbols subgroup Suppose surjective theory topological trivial universal central extension vanishes vector bundles Whitehead