Let $R$ be a $1$-dimensional noetherian local domain. Then we have that $CH^1(R)=\mathbb{Z}/(\gcd([k_i, k]))$ where the $k_i$ are residue fields of the normalization and $k$ is the residue field of $R$.

This was noted in a book, but cannot figure out how to prove, or where it comes from. If anyone could give or sketch a proof, I would be much obliged.