Noncommutative Geometry and Topology in Prague

Principal differential calculi over projective bases

In noncommutative geometry Hopf–Galois extensions generalize the concept of principal bundle. Given a covariant first order differential calculus on the total space algebra there are natural notions of base forms and horizontal forms. If the covariant calculus on the structure Hopf algebra is compatible with the former, in the sense that the vertical map constitutes a short exact sequence, we have a principal differential calculus. Then, the faithfully flat Hopf–Galois extension amplifies to a Hopf–Galois extensions of graded objects. In this talk we give a sheaf-theoretic approach to principal differential calculi. This is required in the presence of projective bases, where global sections are trivial and the local information is crucial. The theory is elucidated by the explicit example of q-deformed SL(2,\mathbb{C}) endowed with its covariant calculi.