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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.