标题： K理论与循环上同调简介 显示英文标题

标题： An Introduction to K-theory and Cyclic Cohomology

摘要： 这些讲义包含K-理论和循环上同调的基本思想的阐述。 我首先列出了一些各种情况下Grothendieck的K函子自然出现的例子，包括拓扑和代数K-理论、C^*-代数的K-理论以及K-同调的初步知识。 然后，我使用代数上的非交换微分形式的Cuntz-Quillen版本，讨论循环上同调的某些基本性质。 作为两种理论之间关系的一个例子，我们描述了Chern同态以及各种类型的指标定理陈述。 讲义的其余部分包含了一些关于循环上同调和约化循环上同调的更详细计算。 这一部分中的关键工具是Goodwillie关于半直积代数的循环复形的定理。 最后一章从Cuntz代数上的超迹的角度，阐述了巴拿赫代数的整个循环上同调。 这里讨论的结果包括整个循环上同调的单纯化归一化、同伦不变性以及导子的作用。 显示英文摘要

摘要： These lecture notes contain an exposition of basic ideas of K-theory and cyclic cohomology. I begin with a list of examples of various situations in which the K-functor of Grothendieck appears naturally, including the rudiments of the topological and algebraic K-theory, K-theory of C^*-algebras, and K-homology. I then discuss elementary properties of cyclic cohomology using the Cuntz-Quillen version of the calculus of noncommutative differential forms on an algebra. As an example of the relation between the two theories we describe the Chern homomorphism and various index-theorem type statements. The remainder of the notes contains some more detailed calculations in cyclic and reduced cyclic cohomology. A key tool in this part is Goodwillie's theorem on the cyclic complex of a semi-direct product algebra. The final chapter gives an exposition of the entire cyclic cohomology of Banach algebras from the point of view of supertraces on the Cuntz algebra. The results discussed here include the simplicial normalization of the entire cyclic cohomology, homotopy invariance and the action of derivations.