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Some questions about algebraic groups.

Let $G$ be an affine algebraic group over algebraically closed field $k$. Questions: 1. $G$ is faithfully flat since it is defined over field? 2. Let $H$ be a closed subgroup of $G$, then (as I learnt from some paper) the map $\pi\colon G\to G/H$ is faithfully flat, why? reference? When it is locally trivial, especially when $G$ is linear algebraic group?

Thank you very much!

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    If $k$ is a field, then any non-zero $k$-vector space whatsoever is faithfully flat. In particular, any non-trivial $k$-algebra is faithfully flat.2012-07-20
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    Yes, its faithfully flat over $k$. I think this is the faithfully flat means in question 1. But it is not obvious that $k[G]$ is faithfully flat over $k[G]^H$ in question 2. Thank you for your reply!2012-07-20
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    Have a look at the Book of Springer: Linear algebraic groups. (Springer is indeed the Author, not the publisher). Your claims are prooven there. I don't have the book by my side, so I cannot give you an exact reference.2012-07-20
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    @sebigu Thank you very much! Actually you can download the book from Springerlink (if your institute have brought the database). You may also get it from my dropbox [link](https://www.dropbox.com/sh/vkl3qqfan4suyza/FsLFapKShz) (will remove after 1day for copyright issue). But I still can not find the answer. Could you help me to find it?2012-07-20
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    Sorry, I wasn't online over the weekend. I will look it up this evening, than send you the reference.2012-07-23

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