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I'm trying to evaluate the cohomology groups of the grassmanian, I think $H^{n}(G(d, n), \mathbb{Q}_{l})$ should be trivial for $n$ odd, and that they are $\mathbb{Q}_{l}$ for $n$ even. Is this correct? does anyone know how to show it?

Alessandro
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    One can cheat and use the comparison theorems between etale cohomology and betti cohomology to arrive at the answer. I myself have never computed the cohomology in either of the settings of etale or betti cohomology but I think the computation of betti cohomology of the grassmanian is quite classical. – random123 Sep 28 '19 at 10:41
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    The Betti numbers of the real/complex Grassmannian are described here. Locally, see here. W.C.Kronholm knows what they speak about, yours truly not necessarily so much :-) – Jyrki Lahtonen Sep 30 '19 at 11:44

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