Tate cohomology

Remarks

Equivalently, $\hat H^n(G, M)$ is the cohomology of $\Hom_G(P_\bullet, M)$ for a complete resolution $P_\bullet$. A complete resolution is an exact complex of projective $\Z[G]$-modules indexed by $\Z$ that agrees with a projective resolution of $\Z$ in nonnegative degrees.

Example: let $G$ be cyclic of order $m$ and let $M = \Z$ with trivial action. Then $M^G = M_G = \Z$ and $N$ is multiplication by $m$. So $\hat H^0(G, \Z) = \Z / m\Z$ and $\hat H^{-1}(G, \Z) = \ker(m \colon \Z \to \Z) = 0$. For cyclic $G$, $\hat H^n(G, M) \cong \hat H^{n+2}(G, M)$ for every $n$ and every $M$.

Example: for $M = \Z[G]$, all groups $\hat H^n(G, \Z[G])$ vanish. In degree $0$, both $M^G$ and $N(M)$ equal $\Z \cdot \sum_{g \in G} g$.

Non-example: for an infinite group $G$, the sum $\sum_{g \in G} gm$ is not defined, so the definition needs $G$ finite.