The Stacks project

59.58 Tate's continuous cohomology

Tate's continuous cohomology ([Tate]) is defined by the complex of continuous inhomogeneous cochains. We can define this when $M$ is an arbitrary topological abelian group endowed with a continuous $G$-action. Namely, we consider the complex

where the boundary map is defined for $n \geq 1$ by the rule

and for $n = 0$ sends $m \in M$ to the map $g \mapsto g(m) - m$. We define

Since the terms of the complex involve continuous maps from $G$ and self products of $G$ into the topological module $M$, it is not clear that this turns a short exact sequence of topological modules into a long exact cohomology sequence. Another difficulty is that the category of topological abelian groups isn't an abelian category!

However, a short exact sequence of discrete $G$-modules does give rise to a short exact sequence of complexes of continuous cochains and hence a long exact cohomology sequence of continuous cohomology groups $H^ i_{cont}(G, -)$. Therefore, on the category $\text{Mod}_ G$ of Definition 59.57.1 the functors $H^ i_{cont}(G, M)$ form a cohomological $\delta $-functor as defined in Homology, Section 12.12. Since the cohomology $H^ i(G, M)$ of Definition 59.57.2 is a universal $\delta $-functor (Derived Categories, Lemma 13.16.6) we obtain canonical maps

for $M \in \text{Mod}_ G$. It is known that these maps are isomorphisms when $G$ is an abstract group (i.e., $G$ has the discrete topology) or when $G$ is a profinite group (insert future reference here). If you know an example showing this map is not an isomorphism for a topological group $G$ and $M \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}_ G)$ please email stacks.project@gmail.com.