$G$ is called a $d$-dimensional duality group if there exist a $G$-module $I$ and an element $e\in H_d(G,I)$ such that, for any $G$-module $M$, the cap product with $e$ induces an isomorphism from $H^{k}(G,M)$ to $H_{d-k}(G,M\otimes I)$ for every integer $k$.
Of course, this implies that the cohomological dimension is $d$. Does it also imply that the homological dimension of $G$ is $d$? In other words, does it imply that $H_{i}(G,Q)=0$ for $i>d$ for every module $Q$, not necessarily of the form $I\otimes M$?
