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.



Definition of Chow groups over Spec Z
One can define Chow groups over any Noetherian scheme $X$. Let $Z\_iX$ be the free abelian group on the $i$-dimensional
subvarieties (closed integral subschemes) of $X$. For any $i+1$-dimensional
subscheme $W$ of $X$, and a rational function $f$ on $W$, we can
define an element of $Z\_iX$ as follows:
$$ [div(f,W)] = \sum\_{V} ord\_V(f)[V] $$
summing over all codimension one subvarieties $V$ of $W$. Then the $i$-Chow group $CH\_i(X)$ is defined
as the quotient of $Z\_iX$ by the subgroup generated by all elements of the form $[div(f,W)]$. The order function is defined as length of the corresponding local ring, so it does not need any further assumptions on $X$, In fact, $X$ being locally Noetherian is enough. This definition also agrees with the one in the original question. 


