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?

finitefield, so the Pic^0 is finite, as are the vector spaces showing up. In particular, all Weil divisors on X are torsion. $\endgroup$