On the cohomology of the classifying spaces of projective unitary groups

Let Formula: see text be the classifying space of Formula: see text, the projective unitary group of order Formula: see text, for Formula: see text. We use a Serre spectral sequence to determine the ring structure of Formula: see text up to degree Formula: see text, as well as a family of distinguished elements of Formula: see text, for each prime divisor Formula: see text of Formula: see text. We also study the primitive elements of Formula: see text as a comodule over Formula: see text, where the comodule structure is given by an action of Formula: see text on Formula: see text corresponding to the action of taking the tensor product of a complex line bundle and an Formula: see text-dimensional complex vector bundle.