---
tag: A3ZHAPT767
kind: definition
name: Tate cohomology
msc: [20J06, 11R34]
defines: 1CA6GZG0R6
uses: [CFAG3VRXEA, 0NZ8JA25WK, 0KVGH0J5B3]
sources:
  - url: https://en.wikipedia.org/wiki/Tate_cohomology_group
---

## Definition

Let $G$ be a finite group and let $M$ be a $G$-module, a [[module|0NZ8JA25WK]] over the group ring $\Z[G]$.
Let $M^G$ be the submodule of elements fixed by $G$, and let $M_G = M / I_G M$ be the module of coinvariants, where $I_G M$ is generated by all $gm - m$.
The norm map $N \colon M \to M$ is $N(m) = \sum_{g \in G} gm$; it induces a map $\bar N \colon M_G \to M^G$.
The *Tate cohomology groups* $\hat H^n(G, M)$, for $n \in \Z$, are:

1. $\hat H^n(G, M) = H^n(G, M)$ for $n \ge 1$, the group cohomology.
2. $\hat H^0(G, M) = M^G / N(M)$.
3. $\hat H^{-1}(G, M) = \ker(\bar N \colon M_G \to M^G)$.
4. $\hat H^{-n-1}(G, M) = H_n(G, M)$ for $n \ge 1$, the [[group homology|CFAG3VRXEA]].

## 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|0KVGH0J5B3]] 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.