代數(shù)K理論及其應(yīng)用

內(nèi)容概要

代數(shù)K理論在代數(shù)拓?fù)?、?shù)論、代數(shù)幾何和算子理論等現(xiàn)代數(shù)學(xué)各個(gè)領(lǐng)域中的作用越來越大。這門學(xué)科的廣泛性往往使人感覺望而生畏。本書以1990年秋天Maryland大學(xué)講義為基礎(chǔ)，不僅為數(shù)學(xué)領(lǐng)域研究生提供很好的學(xué)習(xí)代數(shù)K理論的基本知識(shí)，也講述其在各個(gè)領(lǐng)域的應(yīng)用。全書結(jié)構(gòu)完整，了解代數(shù)基礎(chǔ)知識(shí)、基本代數(shù)拓?fù)浜蛶缀瓮負(fù)渲R(shí)就可以完全讀懂這本書。該書也涉及到不少代數(shù)拓?fù)洹⑼負(fù)浯鷶?shù)和代數(shù)數(shù)論的知識(shí)。最后一章簡(jiǎn)明地介紹了循環(huán)同調(diào)以及其與K理論的關(guān)系。目次：環(huán)的K0群；環(huán)的K1群；范疇的K0、K1群，MilnorK2群；QuillenK理論和+-結(jié)構(gòu)；循環(huán)同調(diào)及其與K理論的關(guān)系?！　∽x者對(duì)象：數(shù)學(xué)系高年級(jí)學(xué)生及研究生的教材，也可供高校數(shù)學(xué)教師及數(shù)學(xué)研究人員閱讀或參考。

書籍目錄

Preface Chapter 1. Ko of Rings 　1. Defining K0 　2. Ko from idempotents 　3. Ko of PIDs and local rings 　4. Ko of Dedekind domains 　5. Relative Ko and excision 　6. An application: Swan's Theorem and topological K- theory 　7. Another application: Euler characteristics and the Wall finiteness obstruction Chapter 2. K1 of Rings 　1. Defining K1 　2. K1 of division rings and local rings 　3. K1 of PIDs and Dedekind domains 　4. Whitehead groups and Whitehead torsion 　5. Relative K1 and the exact sequence Chapter 3. Ko and K1 of Categories, Negative K-Theory 　1. Ko and K1 of categories, Go and G1 of rings 　2. The Grothendieck and Bass-Heller-Swan Theorems 　3. Negative K-theory Chapter 4. Milnor's K2 　1. Universal central extensions and H2 　Universal central extensions 　Homology of groups 　2. The Steinberg group 　3. Milnor's K2 　4. Applications of K2 　Computing certain relative K1 groups 　K2 of fields and number theory 　Almost commuting operators 　Pseudo-isotopy Chapter 5. The +-Construction and Quillen K-Theory 　1. An introduction to classifying spaces 　2. Quillen's +-construction and its basic properties 　3. A survey of higher K-theory 　Products 　K-theory of fields and of rings of integers 　The Q-construction and results proved with it 　Applications Chapter 6. Cyclic homology and its relation to K-Theory 　1. Basics of cyclic homology 　Hochschild homology 　Cyclic homology 　Connections with "non-commutative de Rhom theory" 　2. The Chern character 　The classical Chern character 　The Chern character on Ko 　The Chern character on higher K-theory 　3. Some applications 　Non-vanishing of class groups and Whitehead groups 　Idempotents in C*-algebras 　Group rings and assembly maps 　References 　Books and Monographs on Related Areas of Algebra,Analysis, Number Theory, and Topology 　Books and Monographs on Algebraic K-Theory Specialized References Notational Index Subject Index

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