= Poincare duality group
{c}
{title2=$PD^n$}
A Poincare duality group of dimension $n$ has a finite-length resolution of the trivial <group ring> module $\mathbb Z$ by <finitely generated projective modules>, and $H^k(G,\mathbb ZG)$ vanishes except in degree $n$, where it is infinite cyclic as an abelian group. The top module $D$ can have an orientation-sign action. Reversing the dual projective resolution gives $H^k(G,V)\cong\operatorname{Tor}^{\mathbb ZG}_{n-k}(D,V)$, equivalently $H_{n-k}(G,D\otimes V)$ with the appropriate diagonal action. Untwisted duality requires trivial orientation action. In positive dimension $H^0(G,\mathbb ZG)=0$, since a finitely supported invariant element cannot exist for an infinite group.
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