Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}\_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature? 


For example, do we know


(1) $H^i(\mathrm{Sp}\_{2n}(\mathbb{Z}),\mathbb{Z}^{2n})$ for $i=1,2$? 


(2) $H^i(\mathrm{Sp}\_{2n}(\mathbb{C}),\mathbb{C}^{2n})$ for $i=1,2$? 



Cohomology of linear algebraic groups
Let me focus on (1). It is answered in Lemma A.3 of [this paper](https://arxiv.org/abs/1902.10097) by Krannich. However, let me explain why his answers are 2-torsion. This argument also works for (2) in either interpretation.


That $H^i(Sp\_{2n}(\mathbb{Z});\mathbb{Z}^{2n})$ is 2-torsion is equivalent to it being zero after inverting 2 in the coefficients. This can be proven using the "centre kills" trick. The group $Sp\_{2n}(\mathbb{Z})$ has $C\_2 = \{\pm id\}$ in its center, so we can take the quotient $Sp\_{2n}(\mathbb{Z})/C\_2$ and attempt to compute the desired cohomology groups using the Serre spectral sequence with coefficients in $\mathbb{Z}[1/2]^{2n}$ for the fibration sequence


$$BC\_2 \to BSp\_{2n}(\mathbb{Z}) \to B(Sp\_{2n}(\mathbb{Z})/C\_2),$$


But $H^\*(BC\_2;\mathbb{Z}[1/2]^{2n})$ vanishes; by a transfer argument it vanishes in positive degrees and is given by the invariants in degree 0. Thus the $E^2$-page of the spectral sequence vanishes and hence so does its abutment $H^\*(Sp\_{2n}(\mathbb{Z});\mathbb{Z}[1/2]^{2n})$.


If you care about the 2-torsion in higher degrees, it becomes more difficult. I suggest combining homological stability with polynomial coefficients with a variation of work of Djament and Vespa expressing the stable homology with twisted coefficients in terms of functor homology (e.g. [this paper](https://hal.archives-ouvertes.fr/hal-00315573/fr/)).


