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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.

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    $\begingroup$ What book? ${}{}{}{}{}{}{}{}{}{}{}{}$ $\endgroup$
    – Steven Landsburg
    Commented May 13, 2014 at 23:58
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    $\begingroup$ This follows more or less from the definitions: Since $R$ is one dimensional and local the group is generated by the class of the unique closed point. The relations are given (as in Fulton's book) using the normalisation which is a PID (since it is one dimensional normal and semi-local, assuming perhaps that the normalisation is finite). The image in $CH^1(R)$ of the class of a closed point in the normalisation is given by $[k_i:k]$ times the generator where $k_i$ is the corresponding reside field. $\endgroup$
    – naf
    Commented May 14, 2014 at 5:57

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