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Let $K$ be a field and $k<n$. Denote by $G(k,n)$ the Grassmannian of $k$-dimensional vecors subspaces of $K^n$. I'm trying to check that the projection map

$$G(k+1,n)\times G(k,n)\longrightarrow G(k,n)$$ is open.

I need this in order to check that the projection map from the flag variety $\mathcal{F\ell}(K^n;k,k+1)\longrightarrow G(k,n)$ is open.

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1 Answer 1

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Flat morphisms locally of finite presentation are universally open (EGA IV2, Théorème 2.4.6), and thus your morphism is open.

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  • $\begingroup$ Many thanks. Is there a more direct proof? $\endgroup$ Commented Apr 20, 2018 at 0:35
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    $\begingroup$ This is pretty direct? It's basically just an application of Going Down, see for example stacks.math.columbia.edu/tag/00I1 $\endgroup$
    – KReiser
    Commented Apr 20, 2018 at 0:40

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