Let $p:X\to S$ be a morphism of schemes. Let $\mathcal F$ be an $\mathcal O\_X$-modules. Assume that:


* $\mathcal F$ is quasi-coherent of finite type;
* $\mathcal F$ is flat over $S$;
* the support of $\mathcal F$ is proper over $S$.


Let $\varphi\in\mathrm{Aut}\_{\mathcal O\_X}(\mathcal F)$ be an automorphism of $\mathcal F$
$$\varphi:\mathcal F\xrightarrow{\cong}\mathcal F.$$


Consider the following functor
\begin{equation}
F:(\mathrm{Sch}/S)^{\mathrm{opp}}\to \mathrm{Set}
\end{equation}
\begin{equation}
(T\xrightarrow{h} S)\mapsto \begin{cases}\{\*\},&\quad \textrm{if } h^\*\varphi=\mathrm{id}\_{h^\*\mathcal F}\\\emptyset,&\quad \textrm{else}\end{cases}. 
\end{equation}


I expect that the functor $F$ is representable by a subscheme of $S$.


An example is when $X=\mathrm{GL}(n)$ and $S=\mathrm{Spec}(\Bbbk)$ and $\mathcal F=\mathcal O\_X^n$ and $\varphi:\mathcal F\to\mathcal F$ is the tautological automorphism. Then the functor $F$ is representable by the identity $e\hookrightarrow \mathrm{GL}(n)$.



For an automorphism of a flat family of sheaves, is there a subscheme of the base where the automorphism is identity?
First, replacing $X$ by the support of $\mathcal{F}$ we may assume $X$ is proper over $S$. Second, using Chow's lemma we may assume $X$ is projective over $S$ (I assume here that $S$ is noetherian).


Now, the condition $\varphi = \mathrm{id}$ is equivalent to
$$
\varphi - \mathrm{id} = 0.
$$
Let $L$ be a line bundle on $X$ which is ample over $S$. For each integer $N$ and let
$$
\psi\_N \colon p\_\*(\mathcal{F} \otimes L^N) \to p\_\*(\mathcal{F} \otimes L^N)
$$
be the morphism induced by $\varphi - \mathrm{id}$. Note that $p\_\*(\mathcal{F} \otimes L^N)$ is locally free on $S$ by the flatness assumption. Let $Z\_N \subset S$ be the zero locus of $\psi\_N$. Then the functor $F$ is represented by $Z\_N$ for $N \gg 0$.


