Basis for sum of all extensions of a completion: $\bigoplus_{w \mid v} \mathcal{O}_{w}$ over $\mathcal{O}_{v}$

Question

I was going over notes from a class and it was stated (without proof) that if $\xi_{1}, \ldots, \xi_{n}$ is a basis of $K/k$, then for almost all places $v$, $\xi_{1}, \ldots, \xi_{n}$ is a basis for $\bigoplus_{w \mid v} \mathcal{O}_{w}$ as an $\mathcal{O}_{v}$ module. I'm sure there are references out there, but I haven't been able to find one. I would greatly appreciate it if someone could either help me out with a proof or a reference.

Answer 1

This is just the statement that a torsion module over a Dedekind domain localizes to zero at all but finitely many primes.

Concretely, let $u_1, \dots, u_n$ be a basis for $\mathcal{O}_K$ over $\mathcal{O}_k$. Then the matrix $A$ such that $(u_1, \dots, u_n) = (\xi_1, \dots, \xi_n) \cdot A$ has entries in $K$ and non-zero determinant. Hence there is a finite set of primes $S$ such that both $A$ and $A^{-1}$ have entries which are integral at all primes outside $S$; and it follows that $(\xi_1, \dots, \xi_n)$ is a basis of $\mathcal{O}_K \otimes \mathcal{O}_{k, v}$ over $\mathcal{O}_{k, v}$ for all $v \notin S$.

EDIT: More precisely, the determinant argument shows that there is a finite set of primes $S$ such that the set $\{ \xi_1 \otimes 1, \dots, \xi_n \otimes 1\}$ is a basis of $\mathcal{O}_K \otimes \mathcal{O}_k[1/S]$ over $\mathcal{O}_k[1/S]$. Since $\mathcal{O}_{k, v}$ is an $\mathcal{O}_k[1/S]$-algebra for any $v \notin S$ this implies that $\{ \xi_1 \otimes 1, \dots, \xi_n \otimes 1\}$ is a basis of $\mathcal{O}_K \otimes \mathcal{O}_{k, v}$ over $\mathcal{O}_{k, v}$ for any $v \notin S$.