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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?

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 and flat over $Spec \; \mathbb{Z}$? Does something go wrong with the above definition?