  We determine a subring of the Chow ring and the cohomology of $BPGL_n$, the
classifying space of the projective linear group of degree $n$ over complex
numbers, and explain a way in which this computation might play a role in the
period-index problem. In addition, we show that the Chow ring of $BPGL_n$ is
not generated by the Chern classes of linear representations of $PGL_n$.
