Abstract

The aim of these lectures is to study Connes' cyclic cohomology. It consists of two main parts, definitions and basic properties of cyclic cohomology and cyclic cohomology in the context of noncommutative differential geometry. Let $A$ be an algebra. We begin the first part with Hochschild cohomology $HH^*(A,M)$ of A with coefficient in an $A$-bimodule $M$. Then, Connes' cyclic cochain complex $C^*_{\lambda}(A)$ of $A$ appears as a sub-complex of Hochschild cochain complex $C^*(A,A^{*})$. In order to observe the relation between cyclic cohomology and Hochschild cohomology, which is illustrated in Connes' exact sequence, we define cyclic cochain bicomplex $CC^{**}(A)$ of $A$. We reduce $CC^{**}(A)$ to a shorter cochain bicomplex known as $(b,B)$-bicomplex. Periodic and negative cyclic cohomology of $A$ is discussed next. The second part is devoted to Connes' noncommutative differential calculus. We begin this part by introducing universal differential forms. Then, we define cycles over an algebra $A$ and their characters. The main theorem of this part states that cyclic cocycles in cyclic cocomplex of $A$ are exactly the characters of cycles over $A$. Afterwards, we explain briefly how a Fredholm module over an involutive algebra $A$ defines a cycle over $A$. This allows us to define the Connes-Chern character of a Fredholm module over $A$ as a cyclic cocycle in the periodic cyclic cohomology of $A$. The pairing of these cyclic cocycles with elements of K-theory of $A$ is the noncommutative version of index formula.



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