  We show that the cohomology ring of a quiver Grassmannian asssociated with a
rigid quiver representation has property (S): there is no odd cohomology and
the cycle map is an isomorphism; moreover, its Chow ring admits explicit
generators defined over any field. From this we deduce the polynomial point
count property. By restricting the quiver to finite or affine type, we are able
to show a much stronger assertion: namely, that a quiver Grassmannian
associated to an indecomposable (not necessarily rigid) representation admits a
cellular decomposition. As a corollary, we establish a cellular decomposition
for quiver Grassmannians associated with representations with rigid regular
part. Finally, we study the geometry behind the cluster multiplication formula
of Caldero and Keller, providing a new proof of a slightly more general result.
