Timeline for Quotient of a sheaf by group action and representabillity

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May 20, 2018 at 18:49	comment	added	Jason Starr		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.
May 20, 2018 at 18:40	history	edited	john	CC BY-SA 4.0	edited title
May 20, 2018 at 18:33	history	edited	john	CC BY-SA 4.0	edited body
May 20, 2018 at 18:26	history	edited	john	CC BY-SA 4.0	added 48 characters in body
May 20, 2018 at 18:00	history	asked	john	CC BY-SA 4.0