This question is about a technical issue I ran into.


Let $S$ be a connected 1-dimensional Dedekind scheme, and let $X\to S$ be a flat projective integral normal 2-dimensional scheme. (For simplicity, we can also assume that the generic fibre of $X\to S$ is smooth.)


Is every Weil divisor on $X$ a $\mathbf{Q}$-Cartier divisor? That is, does every Weil divisor on $X$ have the property that a certain integer multiple is a Cartier divisor on $X$?


The answer to this question is positive by Lemma 3.3 in Moret-Bailly's *Groupes de Picard et problemes de Skolem,I* if $S$ is affine, excellent and satisfies Condition (T) on page 162 of Moret-Bailly's article. In particular, if $S$ is the spectrum of the ring of integers of a number field the answer to this question is positive. 


I suspect that the answer to the above question is positive for any excellent Dedekind scheme. Is this known? 



Is every Weil divisor on an arithmetic surface Q-Cartier
The property of being a Cartier divisor is a local property, by definition. Similarly, the property of being a Q-Cartier divisor is also local, since one can take the gcd of the relevant $n$s.


Therefore, the statemeent is true for a scheme $S$ if and only if it is true for each affine subscheme. Then, apply the theorem.


