David Rydh: Flatness, projectivity and purity in algebraic geometry

I will give a survey of some results on projective modules of infinite rank due to Kaplansky, Bass and Raynaud-Gruson. In particular, Raynaud-Gruson introduced the notion of purity that gives a geometric characterization of projectivity for affine morphisms. I will explain how purity and projectivity are related for non-affine morphisms. I will also present an application (in the setting of algebraic stacks) related to an old result of Serre on comodules where projectivity, in contrast to flatness, is crucial.