Hochschild, Cyclic and Periodic Cyclic Homology

Hochschild homology (along with cyclic and periodic cyclic homologies) plays in the non-commutative geometry the role which de Rham cohomology plays in the classical geometry. It is defined for any associative algebra. The Hochschild chains over the algebra \(\mathcal {A}\) are not localised and the operations with the chains over the algebra \(\mathcal {A}\) are not commutative. If the algebra were the algebra of differentiable functions over a topological manifold M, the corresponding Hochschild chains would be differentiable functions over MN. Cyclic/periodic cyclic homology of the \(\mathcal {A}\) were introduced to extend the Chern–Weil characteristic classes to idempotents over \(\mathcal {A}\). Cyclic/periodic cyclic homology represents the minimal algebraic structure for which the Chern–Weil construction works. The cyclic/periodic cyclic homology of the algebra of differentiable functions constitutes the link between the classical differential geometry and non-commutative geometry.