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Let $k$ be a field (algebraically closed if you will) and $G$ be a connected reductive group over $k$. I would like to know a purely algebraic computation of the cohomology of $BG$, as the quotient stack $[\mathrm{pt}/G]$. When $k$ is of characteristic $p$, by cohomology, I mean $\ell$-adic cohomology (i.e. with coefficient in $\overline{\mathbb{Q}}_\ell$, $\ell \neq p$).

Thank you.

PS. I seem to lack the following ingredient to complete the computation. Namely, I need the map $H^*(BT) \to H^*(G/T)$, induced from the obvious map $G/T \to BT$, to be surjective. This will allow me to deduce that the Leray-Serre spectral sequence for $BT \to BG$ degenerates. Surely this is a known result, but I can't find a proof that can be adapted to the $\ell$-adic setting.

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    $\begingroup$ This is proved by Grothendieck in his work "Torsion homologique et section rationelles", Corollair 4, see archive.numdam.org/ARCHIVE/SCC/SCC_1958__3_/SCC_1958__3__A5_0/… He uses Chow rings instead of cohomology but this doesn't matter because $G/B$ has a filtration by affine cells, and so its Chow motive is a sum of Lefschetz motives, and the same decomposition holds for any reasonable cohomology theory. $\endgroup$
    – Victor Petrov
    Commented Apr 3, 2014 at 22:13
  • $\begingroup$ @VictorPetrov: That looks good. Thank you. I will take a look. $\endgroup$
    – QcH
    Commented Apr 4, 2014 at 19:26
  • $\begingroup$ The ideas in arxiv.org/abs/1108.5351 section 7.2 might generalise to characteristic $p$ $\endgroup$
    – Pulcinella
    Commented Feb 11, 2023 at 21:37

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