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On homology of linear groups over $$k[t]$$. (English) Zbl 1322.20037
The author claims to prove the following theorem. Let $$k$$ be an infinite field and let $$G$$ be a connected reductive smooth linear algebraic group over $$k$$. Then the canonical inclusion $$k\hookrightarrow k[t]$$ induces isomorphisms $$H_\bullet(G(k),\mathbb Z)\to H_\bullet (G(k[t]),\mathbb Z)$$, if the order of the fundamental group of $$G$$ is invertible in $$k$$.
He does not explain what is meant by the fundamental group for a nonsplit $$G$$.
In the proof of the first proposition he claims without explanation that for a reductive $$G$$ (over $$k$$) the abelianization map $$G\to G/(G,G)$$ splits (over $$k$$). This deserves a reference or a proof.
In the next proposition he claims without reference the existence and structure of a cover $$\widetilde G$$ of a semisimple, not necessarily split $$G$$. Now it gets confusing. The factors of $$\widetilde G$$ are described as almost simple and simply-connected, but in the summary of the section and in the sequel it seems he believes they are absolutely almost simple and simply-connected.
He truly loses me when he asks to fix a choice of Borel subgroup in $$G(k)$$ containing $$T(k)$$. What does this mean? Say $$T(k)$$ is a circle group in $$G(k)=\mathrm{SO}_3(\mathbb R)$$.
He cites work of Margaux as applying to isotropic simply-connected absolutely almost simple groups and then blithely ignores the isotropy condition.

##### MSC:
 20G10 Cohomology theory for linear algebraic groups 20G35 Linear algebraic groups over adèles and other rings and schemes
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