The Cohomology of Pro-p-Groups with Group Ring Coefficients and Virtual Poincare Duality

The relationship between the group-theoretic properties of a pro-p-group G and the G-module structure of the group $H^n (G,mathbb{F}_q left[kern-0.15emleft[ G right]kern-0.15emright])$ is studied. A necessary and sufficient condition for a pro-p-group G to contain an open Poincare subgroup of dimension n is obtained. This condition does not require that G have finite cohomological dimension and, therefore, applies to groups with torsion. Results concerning the possible values of $dim _{mathbb{F}p} H^n (G,mathbb{F}_p left[kern-0.15emleft[ G right]kern-0.15emright])$ are also obtained.