Picard Group (awaiting review)

Let $S$ be an order in an étale algebra $K$ over $\Q$. The Picard group $\Pic(S)$ of $S$ is the group of invertible fractional $S$-ideals modulo $S$-linear isomorphism. The group operation is ideal multiplication.

The Picard group $\Pic(\mathcal{O}_K)$ of the ring of integers $\mathcal{O}_K$ of a number field $K$ is also referred to as the ideal class group of $K$. Its size is the class number of $K$.