Algebraic K-Theory and Its Applications
Springer Science & Business Media, 6 בדצמ׳ 2012 - 394 עמודים
Algebraic 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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acyclic algebra assembly map basepoint BGL(R central extension chain complex Chern character classifying space cohomology commutative diagram commutative ring compact construction Corollary corresponding CW-complex cyclic group cyclic homology Dedekind domain defined Definition dimension direct sum double complex element example Exercise fact fibration field follows free R-module functor fundamental group gives GL(n GL(R HCo(R hence Ho(G Hochschild homology homology groups homomorphism homotopy class homotopy equivalence Hurewicz map ideal idempotent identity inclusion induces an isomorphism injective integers inverse k-algebra K-groups kernel Ki(R Ko(R Lemma Let G long exact sequence matrix Mn(R morphism multiplication Noetherian non-trivial non-zero Note operator path-connected polynomial prime projective modules proof of Theorem Proposition Quillen quotient map R-module ring and let short exact sequence SL(n split St(R Steinberg symbol subgroup Suppose surjective tensor theory Tn(X topological trivial vanishes vector bundle Whitehead