Consider topological group $\mathbb{R}^\times$ with a standard topology and $\mathbb{R}^\times$-module $S'(\mathbb{R})$ consisting of all Schwartz distributions on $\mathbb{R}$ where $\alpha\in\mathbb{R}^\times$ acts by change of variable: $u(x) \mapsto u(\alpha x)$. The module $S'(\mathbb{R})$ is an infinite-dimensional topological vector space over $\mathbb{C}$.

The $0$'th cohomology group $H^0(\mathbb{R}^\times,S'(\mathbb{R}))$ is one-dimensional, and consists of constant functions, which are distributions invariant under the change of variable.

What about higher $H^k(\mathbb{R}^\times,S'(\mathbb{R}))$? Are they trivial? Finite dimensional? How one can calculate them?

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    $\begingroup$ What definition of group cohomology are you using here? $\endgroup$ – Bruno Joyal Sep 8 '15 at 1:50
  • $\begingroup$ @Bruno Joyal: At first, I thought of the usual definition that uses multivariate functions on $G^n$ (the group $G=\mathbb{R}^\times$ in the the question), with extra continuity requirement, and the usual coboundary conditions. But the answer at mathoverflow.net/questions/72810/… made me doubt, if this is the right definition. $\endgroup$ – Yauhen Radyna Sep 13 '15 at 21:46

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