Assume that $A,I$ is an henselian pair over $R$ and $X$ is a smooth $R$ scheme. can we say that $X(A)\to X(A/I)$ is surjective? I know that this is true if $X$ is affine(or even quasi-projective) or if $(A,I)$ is local henselian But I'm not sure about the general case?
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$\begingroup$ If X is quasi-separated, then this is true (there is a proof in Prop A.0.4 in arxiv.org/pdf/1908.02162.pdf). I suspect quasi-separatedness might be necessary but I don't really know. $\endgroup$– Marc HoyoisCommented Feb 19 at 20:19
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1$\begingroup$ For the general case, see Proposition 6.1.1 here (published in Acta Math. Vietnamica). $\endgroup$– Laurent Moret-BaillyCommented Feb 20 at 16:47
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