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Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus S$. The way I want to define quotient is $ (G\setminus S)(U) = G(U) \setminus S(U) $? Is it necessary that $(G\setminus S)$ is a sheaf? To define $(G\setminus S)$ is necessary to do specification?

Now, suppose $(G\setminus S)$ is representable. Does that imply $S$ is representable?

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    $\begingroup$ There are cases where the quotient presheaf of $S$ by $G$ is a representable sheaf, yet $S$ is not representable. For instance, let $G_0$ be a nontrivial finite group, let $X$ equal $\text{Spec}\ \mathbb{C}[[t]]$ with its closed point $0:\text{Spec}\ \mathbb{C}\to X$, and let $S$ be the fppf sheaf on the category of $X$-schemes associating to every $X$-scheme $Y$ the set of locally constant functions from the closed fiber $Y_0$ to $G_0$. Let $G$ be the constant 'etale group $X$-scheme with $G(X)=G_0$. For the natural action of $G$ on $S$, the quotient is $X$. Yet $S$ is not representable. $\endgroup$
    – Jason Starr
    Commented May 20, 2018 at 18:49

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