Question:
Let $F$ be a finite extension over $\def\q{\mathbb Q}\q$. Let $\mathcal O_F$ be the integral closure of $\mathbb Z$ in $F$. Then if I am not mistaken, a line bundle (an invertible sheaf) over $X:=\operatorname{Spec}(\mathcal O_F)$ is a fractional ideal of $\mathcal O_F$. Let $\mathcal L$ be such a line bundle. Then what is $c_1(\mathcal L)\cap[X]$ as defined in stacks project 02SJ?
My thoughts:
Since the dimension of $X$ is $1$, this should be a $0$-cycle, i.e. a finite linear combination of points of $\operatorname{Spec}(\mathcal O_F)$, modulo principal divisors, namely, an element of the ideal class group of $\mathcal O_F$.
Write $\mathcal L$, regarded as a fractional ideal as $\prod_i\mathfrak p_i^{n_i},\,n_i\in\mathbb Z\,\forall i$, where $\mathfrak p_i$ are prime ideals of $\operatorname{Spec}(\mathcal O_F)$. Then by definition $c_1(\mathcal L)\cap[X]$ is represented by $\sum_i\text{ord}_{\mathfrak p_i, \mathcal{L}}(s) [\mathfrak p_i]$ for some $s\in\mathcal L$. Here $\text{ord}_{\mathfrak p_i, \mathcal{L}}(s)$ is defined as follows.
Take an $s_i\in\mathfrak p_i$ such that $(s_i)=\mathfrak p_i\mathcal O_{F,\mathfrak p_i}$. Then $s=s_i^{n_i}\cdot t_i$ for some $t_i\in\mathcal O_{F,\mathfrak p_i}$. And $\text{ord}_{\mathfrak p_i, \mathcal{L}}(s):=\text{length}(\mathcal O_{F,\mathfrak p_i}/t_i\mathcal O_{F,\mathfrak p_i})$. I think this is equal to $\text{ord}_{p_i}(t_i)$.
So if $\mathcal O_F$ is a P.I.D., then we can choose $s$ such that each $t_i$ is a unit in $\mathcal O_{F,\mathfrak p_i}$, and hence $c_1(\mathcal L)\cap[X]=0,\,\forall\mathcal L$.
In general, I have no idea what do the orders of these $t_i$ mean, in terms of $\mathcal L$.
If I might guess, then I would surmise that the Chern class of a line bundle is just the class of the fractional ideal in the ideal class group. But I do not know if this is correct.
Any references or hints are welcomed. If the above (strange) argument is flawed, please point the flaws out. Thanks in advance.