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?

injective, integral, radiciel, so WLOG (by limits)finiteand can focus onseptdetale objects (good for ZMT). Now $X$ is str. hens. whenever $S$ is, so full faithfulness is seen via ZMT. Essential surj. follows forfiniteetale schemes by etale descent, then in general via stratification (tricky). $\endgroup$ – BCnrd Nov 24 '10 at 13:442more comments