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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.

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