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Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as an $S$-scheme.

Concern: I'm wondering under which "reasonable/ natural" (= not too exotic) conditions this isom scheme $\operatorname{Isom}_S(X,Y)$ is étale over $S$.

The motivation is coined by the comments below Daniel Litt's answer to Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?, where the focus lay on étaleness of $\operatorname{Isom}_{\mathbb{G}_m}(E[\ell], (\mathbb{Z}/\ell\mathbb{Z})^2)$ where the $\ell$ torsion $E[\ell]$ of an elliptic curve $E/\mathbb{G}_m$ is assumed to be étale over $ \mathbb{G}_m$.

And this encourages my curiosity firstly to check which techniques are involved in the last statement's proof and how far it can be generalized. I.e., is it something special for torsion groups of elliptic curves, or broader that the involved objects are group schemes or the rationality of the considered base $\mathbb{G}_m$?
Or can the étaleness of isom scheme above be broadly (how far?) generalized dropping probably some of the properties enumerated in the previous sentence the concrete objects in the linked thread share?

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  • $\begingroup$ When you refer generally to the comments, do you mean specifically this one? $\endgroup$
    – LSpice
    Commented Jan 5 at 15:48
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    $\begingroup$ The Isom scheme exists when $X$ and $Y$ are flat and projective over $S$ (see e.g. FGA Explained, Thm. 5.23), in particular in the finite étale case. For finite étale morphisms, it can also be constructed by descent: a morphism $X\to S$ is finite étale if and only if there is an étale cover $T\to S$ such that $X_T\cong\coprod_{i\in I}T$ for a finite set $I$. If this holds for $X$ and $Y$, choose a cover $T\to S$ trivialising both, so $\mathbf{Isom}_T(X_T,Y_T)$ is also a disjoint union of copies of $T$. Galois descent then constructs $\mathbf{Isom}_S(X,Y)$ as a finite étale scheme over $S$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 5 at 16:04

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