Abstract: I am going to give a basic definition of Hochschild and cyclic cohomology first, and then I will try to give the local index formula in a more general case, if time is enough, I will give the proof of local index formula for the even case.

Abstract: Given the category of modules over an algebra, we would like to know when it is a monoidal category with the usual tensor product of vector spaces. This leads to the notion of Hopf algebra.