Quiver Grassmannians are geometric objects which parametrize subrepresentations of a certain dimension vector of a given representation. We show that for a rigid representation the quiver Grassmannian possesses a diagonal decomposition. This shows that its cohomology is algebraic and also yields a purely algebraic presentation of the cohomology ring in terms of Chern classes of universal bundles. This talk is based on joint work with G. Cerulli Irelli l, F. Esposito, and M. Reineke.