## 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

$C^\bullet _{cont}(G, M) : M \to \text{Maps}_{cont}(G, M) \to \text{Maps}_{cont}(G \times G, M) \to \ldots$

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

\begin{align*} \text{d}(f)(g_1, \ldots , g_{n + 1}) & = g_1(f(g_2, \ldots , g_{n + 1})) \\ & + \sum \nolimits _{j = 1, \ldots , n} (-1)^ jf(g_1, \ldots , g_ jg_{j + 1}, \ldots , g_{n + 1}) \\ & + (-1)^{n + 1}f(g_1, \ldots , g_ n) \end{align*}

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

$H^ i_{cont}(G, M) = H^ i(C^\bullet _{cont}(G, M))$

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

$H^ i(G, M) \longrightarrow H^ i_{cont}(G, M)$

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.

Comment #7013 by Joshua Ruiter on

I believe the assertion regarding isomorphisms when $G$ is profinite is false. A counterexample is given in Gille & Szamuely's Central Simple Algebras and Galois Cohomology remark 4.2.4.

Comment #7014 by on

Are you sure? Because that reference has a different definition for $H^i_{cont}(G, -)$, see Definition 4.2.2 in your reference. Moreover, the definition of $H^i(G, -)$ that is used there is certainly different from what is our $H^i(G, -)$; theirs doesn't use the topology on $G$ and ours does.

In fact, I think that the fact that our $H^i_{cont}$ is equal to their $H^i_{cont}$ is the result that you claim is wrong and that the text above claims is true.

Sorry, but this is just incredibly confusing; so can you please very carefully match the definitions used in both locations and then comment again. Thanks a bunch!

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