代数K理论及其应用

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： Swans 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. Milnors K2

1. Universal central extensions and H2

Universal central extensions

Homology of groups

2. The Steinberg group

3. Milnors 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. Quillens +-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

代数K理论在代数拓扑、数论、代数几何和算子理论等现代数学各个领域中的作用越来越大。这门学科的广泛性往往使人感觉望而生畏。《代数K理论及其应用》以1990年秋天Maryland大学讲义为基础，不仅为数学领域研究生提供很好的学习代数K理论的基本知识，也讲述其在各个领域的应用。全书结构完整，了解代数基础知识、基本代数拓扑和几何拓扑知识就可以完全读懂这《代数K理论及其应用》。该书也涉及到不少代数拓扑、拓扑代数和代数数论的知识。最后一章简明地介绍了循环同调以及其与K理论的关系。目次：环的K0群；环的K1群；范畴的K0、K1群，MilnorK2群；QuillenK理论和+-结构；循环同调及其与K理论的关系。
读者对象：数学系高年级学生及研究生的教材，也可供高校数学教师及数学研究人员阅读或参考。