Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.

Let $h_X$, $h_G$ be the representable sheaves on the small étale site $S_\text{ét}$ associated to $X,G$ respectively. Suppose $h_X$ is a torsor under $h_G$ (i.e., $h_G(T)$ acts simply transitively on $h_X(T)$ for every étale $T\rightarrow S$). Must $X$ be a $G$-torsor? (i.e., $\alpha\times\mathrm{id} : G\times X\rightarrow X\times X$ is an isomorphism).

If necessary, I'm happy to assume $G$ reductive and that $X$ is $S$-smooth and $S$-affine.

If this is true, a pleasing consequence is that it suffices to check that $h_G(\operatorname{Spec} R)$ is simply transitive on $h_X(\operatorname{Spec} R)$ for strict local rings $R$ of $S$.

Remark: At first I thought that maybe smoothness is key to a positive answer (via étale slicing), but now I wonder if the key is affineness, since $S$-affine schemes are the relative specs of quasicoherent modules, which are determined by their restriction to the small étale site….

Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.

Let $h_X$, $h_G$ be the representable sheaves on the small étale site $S_\text{ét}$ associated to $X,G$ respectively. Suppose $h_X$ is a torsor under $h_G$ (i.e., $h_G(T)$ acts simply transitively on $h_X(T)$ for every étale $T\rightarrow S$). Must $X$ be a $G$-torsor? (i.e., $\alpha\times\mathrm{id} : G\times X\rightarrow X\times X$ is an isomorphism).

If necessary, I'm happy to assume $G$ reductive and that $X$ is $S$-smooth and $S$-affine.

If this is true, a pleasing consequence is that it suffices to check that $h_G(\operatorname{Spec} R)$ is simply transitive on $h_X(\operatorname{Spec} R)$ for strict local rings $R$ of $S$.

Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.

Let $h_X$, $h_G$ be the representable sheaves on the small etale site $S_{et}$ associated to $X,G$ respectively. Suppose $h_X$ is a torsor under $h_G$ (i.e., $h_G(T)$ acts simply transitively on $h_X(T)$ for every etale $T\rightarrow S$). Must $X$ be a $G$-torsor? (i.e., $\alpha\times\text{id} : G\times X\rightarrow X\times X$ is an isomorphism).

If necessary, I'm happy to assume $G$ reductive and that $X$ is $S$-smooth and $S$-affine.

If this is true, a pleasing consequence is that it suffices to check that $h_G(\text{Spec }R)$ is simply transitive on $h_X(\text{Spec }R)$ for strict local rings $R$ of $S$.

Remark: At first I thought that maybe smoothness is key to a positive answer (via etale slicing), but now I wonder if the key is affineness, since $S$-affine schemes are the relative specs of quasicoherent modules, which are determined by their restriction to the small etale site....

Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.

Let $h_X$, $h_G$ be the representable sheaves on the small étale site $S_\text{ét}$ associated to $X,G$ respectively. Suppose $h_X$ is a torsor under $h_G$ (i.e., $h_G(T)$ acts simply transitively on $h_X(T)$ for every étale $T\rightarrow S$). Must $X$ be a $G$-torsor? (i.e., $\alpha\times\mathrm{id} : G\times X\rightarrow X\times X$ is an isomorphism).

If necessary, I'm happy to assume $G$ reductive and that $X$ is $S$-smooth and $S$-affine.

If this is true, a pleasing consequence is that it suffices to check that $h_G(\operatorname{Spec} R)$ is simply transitive on $h_X(\operatorname{Spec} R)$ for strict local rings $R$ of $S$.

Remark: At first I thought that maybe smoothness is key to a positive answer (via étale slicing), but now I wonder if the key is affineness, since $S$-affine schemes are the relative specs of quasicoherent modules, which are determined by their restriction to the small étale site….

Let $S$ be a scheme, $G$ a smooth affine group scheme over $S$, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.

Let $h_X$, $h_G$ be the representable sheaves on the small etale site $S_{et}$ associated to $X,G$ respectively. Suppose $h_X$ is a torsor under $h_G$ (i.e., $h_G(T)$ acts simply transitively on $h_X(T)$ for every etale $T\rightarrow S$). Must $X$ be a $G$-torsor? (i.e., $\alpha\times\text{id} : G\times X\rightarrow X\times X$ is an isomorphism).

I'm happy to assume $G$ reductive if it helps.

If this is true, a pleasing consequence is that it suffices to check that $h_G(\text{Spec }R)$ is simply transitive on $h_X(\text{Spec }R)$ for strict local rings $R$ of $S$.

Let $S$ be a scheme, $G$ a smooth $S$-affine $S$-group scheme, and $X$ an $S$-scheme with a ($S$-linear) $G$-action $\alpha : G\times X\rightarrow X$.

Let $h_X$, $h_G$ be the representable sheaves on the small etale site $S_{et}$ associated to $X,G$ respectively. Suppose $h_X$ is a torsor under $h_G$ (i.e., $h_G(T)$ acts simply transitively on $h_X(T)$ for every etale $T\rightarrow S$). Must $X$ be a $G$-torsor? (i.e., $\alpha\times\text{id} : G\times X\rightarrow X\times X$ is an isomorphism).

If necessary, I'm happy to assume $G$ reductive and that $X$ is $S$-smooth and $S$-affine.

If this is true, a pleasing consequence is that it suffices to check that $h_G(\text{Spec }R)$ is simply transitive on $h_X(\text{Spec }R)$ for strict local rings $R$ of $S$.

Remark: At first I thought that maybe smoothness is key to a positive answer (via etale slicing), but now I wonder if the key is affineness, since $S$-affine schemes are the relative specs of quasicoherent modules, which are determined by their restriction to the small etale site....