Let $f:X\to S$ be a universal homeomorphism of schemes. Assume $X(S')\neq\emptyset$ for some étale surjective $S'\to S$. Does $f$ have a section?


The answer is yes if $S$ is reduced, by descent. Indeed, note that if $S\_1$ is a reduced $S$-scheme then $X(S\_1)$ has at most one element. Apply this to $S\_1=S'\times\_S S'$.


Interesting special case: if $S$ has prime characteristic $p$, let $G$ be a finite locally free $S$-group scheme with connected (i.e. "infinitesimal") fibers, such as $\alpha\_p$ or $\mu\_p$. Is $H^1\_{\mathrm{et}}(S,G)$ trivial?



Universal homeomorphisms and the étale topology
I think that the following might work. Let $X\_0$ be a reduced scheme over a field $k$ of characteristic $p > 0$, and let $X$ be the product of $X\_0$ with the ring of dual numbers $k[\epsilon]$. Then $\mathcal O\_X = \mathcal O\_{X\_0} \oplus \epsilon\mathcal O\_{X\_0}$, and the $p^{\rm th}$ roots of 1 are those of the form $1 + \epsilon f$; hence the Zariski sheaf of $p^{\rm th}$ roots on 1 on $X$ is isomorphic to $\mathcal O\_{X\_0}$. Hence if $\mathrm H^1(X\_0, \mathcal O\_{X\_0}) ≠ 0$ there is a non-trivial $\mu\_p$-torsor on $X$ that is locally trivial in the Zariski topology, thus giving a counterexample.


