3 Summary; deleted 1 characters in body

As explained by Hailong, one can define the Chow group CH($X$) (without grading by the dimension of the cycles) of any Noetherian scheme $X$. But in general one has to be careful about the behavior of the rational equivalence.

• A principal divisor in an integral closed subscheme $W$ of dimension $i+1$ is not necessary a $i$-cycle (this has to do with the notion of catenary schemes).

• If $X$ is not equidmensional, then the divisor associated to an invertible rational function can be not rationally equivalent to $0$ (even when $X$ is a reduced projective variety over a field).

• If $f : X\to Y$ is a proper morphism, then contrarily to the case of varieties over a field, the pushforward by $f$ does not induce a map ${\rm CH}(X)\to {\rm CH}(Y)$. One can construct an example with $X$ affine, regular of dimension $2$, $f$ finite birational and the image of a principal divisor on $X$ is non-zero in CH($Y$).

To remedy to these pathologies, Thorup (Rational equivalence theory on arbitrary Noetherian schemes, Enumerative geometry (Sitges, 1987), 256--297, Lecture Notes in Math., 1436, 1990) defines a graded rational equivalence relation associated to a grading on $X$ (a map from $X$ to $\mathbb Z$ with some properties, e.g. $x\mapsto -\dim O_{X,x}$), and a graded principal divisor on an integral closed subscheme is the usual principal divisor where we discount components of bad gradings. With the new rational equivalence everything works fine for the pushforward by proper morphisms $X\to Y$ (though the gradings on $X$ and $Y$ must be compatible in some sense) and pullback by flat morphisms.

When $X$ and $Y$ are of finite type over an universally catenary base scheme $S$ [EDIT which is equidimensional at every point], then any proper morphism $f : X\to Y$ induces a homomorphism ${\rm CH}(X)\to {\rm CH}(Y)$ (without grading), see S. Kleiman, Intersection theory and enumerative geometry: a decade in review, With the collaboration of Anders Thorup on § 3. Proc. Sympos. Pure Math., 46, Part 2, Algebraic geometry, Bowdoin, 1985, 321--370, Amer. Math. Soc., Providence, RI, 1987.

I learned most of these from O. Gabber and the examples mentionned mentioned above are in a (not yet finished) preprint with him and D. Lorenzini.

[EDIT]. To summarize, if $X$ is noetherian, universally catenary (e.g. finite type over ${\mathbb Z}$ or any noetherian regular scheme) and equidimensional at every point (i.e. for every $x\in X$, the irreducible components of ${\rm Spec}(O_{X,x})$ all have the same dimension), then CH$(X)$ can be decomposed as the direct sum of the $CH_i(X)$'s as in Hailong's post. If $f : X\to Y$ is a proper morphism of universally catenary noetherian schemes, then $f_*$ induces a homomorphism ${\rm CH}(X)\to {\rm CH}(Y)$. In the last counterexample above, $X$ is regular (so univ. catenary), $Y$ is catenary but not universally.

2 Forget an important hypothesis.

As explained by Hailong, one can define the Chow group CH($X$) (without grading by the dimension of the cycles) of any Noetherian scheme $X$. But in general one has to be careful about the behavior of the rational equivalence.

• A principal divisor in an integral closed subscheme $W$ of dimension $i+1$ is not necessary a $i$-cycle (this has to do with the notion of catenary schemes).

• If $X$ is not equidmensional, then the divisor associated to an invertible rational function can be not rationally equivalent to $0$ (even when $X$ is a reduced projective variety over a field).

• If $f : X\to Y$ is a proper morphism, then contrarily to the case of varieties over a field, the pushforward by $f$ does not induce a map ${\rm CH}(X)\to {\rm CH}(Y)$. One can construct an example with $X$ affine, regular of dimension $2$, $f$ finite birational and the image of a principal divisor on $X$ is non-zero in CH($Y$).

To remedy to these pathologies, Thorup (Rational equivalence theory on arbitrary Noetherian schemes, Enumerative geometry (Sitges, 1987), 256--297, Lecture Notes in Math., 1436, 1990) defines a graded rational equivalence relation associated to a grading on $X$ (a map from $X$ to $\mathbb Z$ with some properties, e.g. $x\mapsto -\dim O_{X,x}$), and a graded principal divisor on an integral closed subscheme is the usual principal divisor where we discount components of bad gradings. With the new rational equivalence everything works fine for the pushforward by proper morphisms $X\to Y$ (though the gradings on $X$ and $Y$ must be compatible in some sense) and pullback by flat morphisms.

When $X$ and $Y$ are of finite type over an universally catenary base scheme $S$, S$[EDIT which is equidimensional at every point], then any proper morphism$f : X\to Y$induces a homomorphism${\rm CH}(X)\to {\rm CH}(Y)$(without grading), see S. Kleiman, Intersection theory and enumerative geometry: a decade in review, With the collaboration of Anders Thorup on § 3. Proc. Sympos. Pure Math., 46, Part 2, Algebraic geometry, Bowdoin, 1985, 321--370, Amer. Math. Soc., Providence, RI, 1987. I learned most of these from O. Gabber and the examples mentionned above are in a (not yet finished) preprint with him and D. Lorenzini. 1 As explained by Hailong, one can define the Chow group CH($X$) (without grading by the dimension of the cycles) of any Noetherian scheme$X$. But in general one has to be careful about the behavior of the rational equivalence. • A principal divisor in an integral closed subscheme$W$of dimension$i+1$is not necessary a$i$-cycle (this has to do with the notion of catenary schemes). • If$X$is not equidmensional, then the divisor associated to an invertible rational function can be not rationally equivalent to$0$(even when$X$is a reduced projective variety over a field). • If$f : X\to Y$is a proper morphism, then contrarily to the case of varieties over a field, the pushforward by$f$does not induce a map${\rm CH}(X)\to {\rm CH}(Y)$. One can construct an example with$X$affine, regular of dimension$2$,$f$finite birational and the image of a principal divisor on$X$is non-zero in CH($Y$). To remedy to these pathologies, Thorup (Rational equivalence theory on arbitrary Noetherian schemes, Enumerative geometry (Sitges, 1987), 256--297, Lecture Notes in Math., 1436, 1990) defines a graded rational equivalence relation associated to a grading on$X$(a map from$X$to$\mathbb Z$with some properties, e.g.$x\mapsto -\dim O_{X,x}$), and a graded principal divisor on an integral closed subscheme is the usual principal divisor where we discount components of bad gradings. With the new rational equivalence everything works fine for the pushforward by proper morphisms$X\to Y$(though the gradings on$X$and$Y$must be compatible in some sense) and pullback by flat morphisms. When$X$and$Y$are of finite type over an universally catenary base scheme$S$, then any proper morphism$f : X\to Y$induces a homomorphism${\rm CH}(X)\to {\rm CH}(Y)\$ (without grading), see S. Kleiman, Intersection theory and enumerative geometry: a decade in review, With the collaboration of Anders Thorup on § 3. Proc. Sympos. Pure Math., 46, Part 2, Algebraic geometry, Bowdoin, 1985, 321--370, Amer. Math. Soc., Providence, RI, 1987.

I learned most of these from O. Gabber and the examples mentionned above are in a (not yet finished) preprint with him and D. Lorenzini.