Algebra Seminar

Coalgebra-Galois extensions of noncommutative algebras are algebraic analogues of principal bundles. To any such an extension one can associate modules much as vector bundles are associated to principal bundles. We show the equivalence of the relative projectivity of coaugmented coalgebra-Galois extensions and existence of strong connections. Then we prove that modules associated to such extensions via finite dimensional corepresentations are finetely generated projective, and thus fit the formalism of the Chern-Connes pairing between K-theory and cyclic cohomology. As an example, we use the Chern-Connes pairing to define, and the Noncommutative Index Theorem to compute, the Chern numbers of algebraic line bundles associated to the q-deformed Hopf fibration.