On the cohomology of the classifying spaces of projective unitary groups

Abstract

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.