Trieste, August 26-28 2014

K-homology and graph algebras

On differential smoothness of algebras.

The aim of this lecture is twofold. In the first instance we would like to explain the notion of differential smoothness of algebras. There are many ways in which a noncommutative algebra can be understood as a coordinate algebra of a smooth noncommutative variety. Most commonly, one studies homologically smooth algebras, i.e. algebras that admit a finite resolution by finitely generated and projective bimodules. More demanding is the requirement of the existence of a Poincare-type duality between Hochschild homology and cohomology, which is embodied in the Calabi-Yau condition. Here we propose the notion of differential smoothness which is based on the existence of a differential structure which explicitly displays a Poincare-type duality. To formulate this duality, in addition to the de Rham complex of differential forms one needs the complex of integral forms, which we define. The ideas of differential smoothness are explained on two (classes of) examples: the noncommutative pillow and quantum cones. These are both deformations of singular varieties which become smooth upon quantization and they serve as the illustration of the second point of the talk, namely that deformation can often lead to resolution of singularities.