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Let $\hat{\mathbb G}$ be a formal group over a field $k$, and let $V$ be a finite dimensional algebraic representation of $\hat{\mathbb G}$ (meaning we have fixed a homomorphism of algebraic groups $\rho:\hat{\mathbb G}\to\operatorname{GL}(V)$ over $k$).

I would like to consider the rational cohomology of $\hat{\mathbb G}$ with coefficients in $V$, denoted $H^\ast(\hat{\mathbb G},V)$. By definition, this is the cohomology of the complex whose $p$-cochains are the additive group of regular functions $$f:\overbrace{\hat{\mathbb G}\times\cdots\times\hat{\mathbb G}}^{p\text{ times}}\to V$$ and whose differential is given by the usual formula $$(\delta f)(t_1,\ldots,t_p)=\rho(t_1)f(t_2,\ldots,t_p)-f(t_1t_2,t_3,\ldots,t_p)+f(t_1,t_2t_3,t_4,\ldots,t_p)-\cdots$$ I presume much must be known about the resulting cohomology groups $H^\ast(\hat{\mathbb G},V)$, but I do not know where to look.

I believe I can show that $H^\ast(\hat{\mathbb G}_{\mathrm{m}},k_{\mathrm{std}}^{\otimes n})=0$ for integers $n\ne 0$, where $\hat{\mathbb G}_{\mathrm{m}}$ is the formal multiplicative group and $k_{\mathrm{std}}$ is its standard representation. Is this a known result for which there exists a canonical reference? It looks like I need this result for some specific application, and I'd like to get a feeling for what is known in this area.

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    $\begingroup$ For a flat affine group scheme $G$ over a ring $A$, the "Hochschild cohomology" functors for (functorial) linear representations of $G$ on $A$-modules are the derived functors of "$G$-invariants" (see Lemma B.2.3 of math.stanford.edu/~conrad/papers/luminysga3.pdf for a proof). That proves vanishing of higher Hochschild cohomology of $A$-tori $T$ (on $A$-modules with a linear $T$-action) via exactness of "$T$-invariants" (due to gradings over an etale cover of $A$ splitting $T$). Complete reducibility of $\widehat{\mathbf{G}}_m$-actions over $k$ holds in char. $p$, but not in char. 0... $\endgroup$
    – nfdc23
    Apr 8, 2017 at 2:19
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    $\begingroup$ [The "problem" is that in char. 0, $\widehat{\mathbf{G}}_m \simeq \widehat{\mathbf{G}}_a$.] Note that you're using the notation "$k$" for the field and the cohomological degree. Do you want only ${\rm{char}}(k)>0$? Anyway, a very general functor-based reference on Hochschild cohomology over fields is provided inside the massive tome of Demazure-Gabriel on algebraic groups, but I don't remember offhand if they address the case of formal groups in there (and that book can be a bit daunting to navigate). $\endgroup$
    – nfdc23
    Apr 8, 2017 at 2:27
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    $\begingroup$ Minor pedantic point: since $\widehat{\mathbf{G}}_m$ is not a scheme but only a formal scheme (e.g., the comultiplication involves a completed tensor product at the coordinate ring level), one shouldn't speak of a map $\rho: \widehat{\mathbf{G}}_m \rightarrow {\rm{GL}}(V)$ being one of "algebraic groups" but rather as a map of formal group schemes (where the target is an ordinary group scheme viewed as a formal scheme). $\endgroup$
    – nfdc23
    Apr 8, 2017 at 2:34

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