Divisor on an arithmetic surface and "base change"

Question

Fix a number field $K$ with ring of integers $O_K$; moreover $\sigma:K\to \mathbb C$ is an embedding of fields. Let $X\to\operatorname{Spec} O_K$ be an arithmetic surface ($X$ is regular and projective amd geometrically irreducible) and let $D=\sum_Yv_Y(D)Y$ be a divisor in $X$.

I was wondering what is the object usually indicated in the literature with the symbol with $D_\sigma$ (see for example Moriwaki's book about Arakelov geometry at page 101, line -6).

Here my guess: Usually $X_\sigma$ is the Riemann surface defined as the base change $X\times_{\operatorname{Spec} O_K}^\sigma \operatorname{Spec} \mathbb C$ through the morpshism $\operatorname{Spec}(\sigma):\operatorname{Spec} \mathbb C\to\operatorname{Spec} O_K$.The pullback of $D$ through the projection $p: X_\sigma\to X$ is a well defined divisor, since $p$ is flat. Therefore we put $D_\sigma:=p^\ast D$.

Is this argument correct?

Answer 1

We have the following two point of views of divisors on a nonsingular variety $X$.

1. Weil divisor: linear combination of prime divisors.
2. Cartier divisor: an element of $H^0(X, \text{Rat}(X)^\times/\mathcal{O}_X^\times)$, that is, data $\{(U_i, f_i)\}_{i = 1, \ldots, n}$ with the following properties.
   • $\bigcup U_i = X$ and $f_i$ is a rational function on $U_i$.
   • There is a unit function $u_{ij}$ on $U_i \cap U_j$ such that $f_i = u_{ij} f_j$ on $U_i \cap U_j$.

The two point of views are equivalent. The second point of view is better for understanding your question. $D_{\sigma}$ is just the pullback of $D$ by the morphism $X_{\sigma} \to X$.