## homogenous bundles

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Let $G$ be a reductive algebraic group over $\mathbb{C}$ and $H$ be an algebraic subgroup of $G$. We suppose that $H$ acts on some scheme $S$, where $S$ is of finite type over $\mathbb{C}$. Then I would like to know what are the usual assumptions on $H$ and $S$ to have the existence of the scheme $$G \times^{H} S:=(G \times S)/rel$$ where $\forall g \in G, h \in H, s \in S$, we define the relation $rel$ by $$(g,s) \ rel \ (gh^{-1},h.s).$$ It is something well-known in the case where $S$ is an $H$-module, but for $S$ an arbitrary $H$-scheme I have been heard that it dosen't always exist. I would like to have references or precise response about the hypotheses I should do on $S$ and $H$. For instance, I think it's ok if $P$ is a parabolic subgroup of $G$ or even if it is a reductive subgroup but still in those cases I know no reference.

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$S$ doesn't need to have the specific form $\mathbf{V}(M)$ for an $H$-module $M$ - it could be any quasi-projective scheme with $H$-action.  On the other hand, if $H =\mathbb{Z}/2\mathbb{Z}$, there is a counterexample in mathoverflow.net/questions/72152/…  – S. Carnahan♦ Apr 27 2012 at 3:16

## 1 Answer

Section I.5 ("Quotients and associated sheaves") of Jantzen's Representations of Algebraic Groups is (at least in my mind) a standard resource for this question. (Here is a Google books link). He considers your question in full generality there. In particular, he proves (cf I.5.6.(8)) that if $G$ is an algebraic group over a field $k$ and $H$ is a closed subgroup scheme of $G$ then $G/H$ is a scheme. (Here the definition of $G/H$ agrees with what you think it should mean over a field, but in general the definition of $G/H$ is given categorically, cf the definition of the quotient faisceau $X/G$ for any $G$-space $X$ in I.5.5).

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