A noncommutative sheaf theory
Seminar Room 1, Newton Institute
Modules with flat connection over algebras with differential structure have several properties in common with sheaf theory over topological spaces. In particular they admit long exact sequences for a cohomology theory. However sheaf theory is best described by looking at its applications, one of which is the Serre spectral sequence for a topological fibration. In the case where the cohomology of the fiber is not a trivial bundle over the base, sheaf cohomology is required to make sense of the resulting cohomology theory. I will describe a noncommutative version of the Serre spectral sequence for de Rham cohomology, which uses flat connections. This will effectively specify a definition of differential fibration in noncommutative geometry. I will then consider examples, which are quantum homogenous spaces.