Question: 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, 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.

Answer: 
The last chapter of Fulton's book defines Chow groups on schemes $X$ of finite type over a fixed smooth scheme $S$. It not automatic to have an intersection product for $X$ smooth, but there is one as long as $S$ is $1$-dimensional. So it should cover the case of $\mathbb{Z}$.
I think intersection of vertical divisors cannot be defined, see Liu 9.1.33. This should be fixed by Arakelov theory.