If $S$ is a Henselian local scheme with closed point $v$ and $X$ a smooth $S$-scheme, then it is well known that the canonical map $X(S) \rightarrow X(v)$ is surjective.

Suppose now that $Y$ is smooth over $X$. Is the canonical map $X(Y) \rightarrow X(Y_v)$ surjective (all maps are $S$-maps)? I am happy to make additional assumptions on $X$ and $S$, though preferably not too many on $Y$... I am mainly interested in the case $S = Spec(\mathbb{Z}_p)$ or finite extensions.

Thanks,

David

If $S$ is a Henselian local scheme with closed point $v$ and $X$ a smooth $S$-scheme, then it is well known that the canonical map $X(S) \rightarrow X(v)$ is surjective.

Suppose now that $Y$ is smooth over $X$. Is the canonical map $X(Y) \rightarrow X(Y_v)$ surjective (all maps are $S$-maps)? I am happy to make additional assumptions on $X$ and $S$, though preferably not too many on $Y$... I am mainly interested in the case $S = Spec(\mathbb{Z}_p)$ or finite extensions.

Thanks,

David

edit: oops, for the map $X(Y) \rightarrow X(Y_v)$ to make sense I need to assume a bit more about $S$, I think I also need $Y/S$ proper.