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?

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    Better: $f^{\ast}$ is equivalence of etale sites (apply to $S$-etale $S'$ and $S$); a footnote w/o details in 1970 ed. of SGA1. Proof: By EGA IV$_4$, 18.12.11, univ. homeo. = integral, radiciel, & surj. So WLOG $S$ and $X$ affine and (by nilimmersion case: 18.1.2!) reduced. Then ring map is injective, integral, radiciel, so WLOG (by limits) finite and can focus on septd etale objects (good for ZMT). Now $X$ is str. hens. whenever $S$ is, so full faithfulness is seen via ZMT. Essential surj. follows for finite etale schemes by etale descent, then in general via stratification (tricky). – BCnrd Nov 24 '10 at 13:44
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    Dear Brian, does your comment answer Laurent's question? I may be confused (this happens a lot), but I don't see it. – Angelo Nov 24 '10 at 15:47
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    I'm probably being dense, but here's a thought: work over a perfect field k of char p, let X = E x alpha_p, where E is an elliptic curve over k, and let G = alpha_p viewed as a sheaf on X_et. Any etale V -> X is of the form U x alpha_p -> X for some etale U -> E by top. invariance. So if I'm not mistaken, this means the following: under the identification of the small etale topoi of E and X, the sheaf defined by G on X is identified by G_a^{p-1} on E (compute maps k[x]/x^p -> R[x]/x^p with R reduced). In particular, there exist lots of non-trivial etale locally trivial G-torsors on X. – Bhargav Nov 24 '10 at 17:20
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    Bhargav, great. OK for $\mu_p$, not sure for $\alpha_p$. For $G = \mu_p$, finiteness of $f:X = G \times E \rightarrow E$ implies ${\rm{H}}^1(X,F) = {\rm{H}}^1(E,f_{\ast}F)$ for abelian etale $F$ on $X$. Take $F$ to be functor of pts of $G$ on etale $X$-schemes, so $f_{\ast}F$ is functor of pts of ${\rm{R}}_{X/E}(G) = {\rm{R}}_{G/k}(G) \times E$ on etale $E$-schemes (all reduced). By left-exactness, this is $E$-gp obtained from $k$-gp ${\rm{R}}_{G/k}(\mathbf{G}_m)[p] = \mu_p \times (1 + (t))/(1 + (t^p)) = \mu_p \times \mathbf{G}_a^{p-1}$ (use truncated log), underlying reduced is $O_E^{p-1}$. – BCnrd Nov 24 '10 at 18:29
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    Brian, great! So in these two cases all etale locally trivial G-torsors on X are actually Zariski locally trivial. Is that generally true? It seems mildly plausible that some variant of the restriction of scalars argument gives an affirmative answer when the inclusion of reduced subscheme admits a retraction, and so should be okay in equicharacteristic when the reduction is smooth... – Bhargav Nov 24 '10 at 18:44
up vote 6 down vote accepted

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.

  • Angelo, great, that's a really simple version of Bhargav's argument. Thanks to Bhargav and Brian as well! – Laurent Moret-Bailly Dec 3 '10 at 10:36

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