Usually (eg, intro. in M. Rost's 'Cycle modules with coefficients'), for a variety, $X$, over a field one can define the Chow group of p-cycles, $CH_p (X)$, as $$CH_p (X) = coker\; \left[\bigoplus_{x\in X_{p+1}} k(x)^\times \rightarrow \bigoplus_{x\in X_p} \mathbb{Z}\; \right]$$.
What about for an arithmetic scheme, eg when $X$ is, say, normal, separated, finitely generated of finite type and flat over $Spec \; \mathbb{Z} $? Does something go wrong with the above definition?
Peter Arndt had posed part of this question already, but it seems without an answer.
