Expressing $\mathcal O_L$ as a certain free module of rank 1

Question

I have a finite Galois extension of number fields $L/K$ with group $G$ and respective rings of integers $\mathcal O_L$ and $\mathcal O_K$.

If $\Gamma$ is an $\mathcal O_K$-order in $K[G]$ and $\mathcal O_L$ is free as a $\Gamma$-module, then does it always have rank 1?

Many thanks in advance.

Answer 1

The answer is trivially yes, because $K \otimes_{O_K} O_L = L$ is always free of rank 1 as a module over $K \otimes_{O_K} \Gamma = K[G]$.