Existence of induced map on Divisor Class Group? Let $f: X \rightarrow Y$ be a morphism of noetherian, integral schemes, regular in codimension 1 (so we can talk about Weil divisors). I am wondering whether there is an induced map on divisor class groups in either direction. It seems like such a map need not exist in general. So far, I have tried to come up with a natural definition in the affine case and failed. Given a ring map $f^\# : A \rightarrow B$ to get a contravariant map $f^* : Cl(Spec(A)) \rightarrow Cl(Spec(B))$ the natural thing to try was to map a height 1 prime $\mathfrak{p}$ to $f^\#(\mathfrak{p})B =: \mathfrak{q}$. However, of course in general $\mathfrak{q}$ need not be a height one prime of $B$. The next thing I tried was to maybe consider all minimal primes containing $\mathfrak{q}$ and associate to $\mathfrak{p}$ some appropriate linear combination of these. This I guess will not work in general?
There seem to be specific examples in Hartshorne where such induced maps are defined - (the case of finite morphisms of non-singular curves for instance). So are there some reasonable conditions that can be imposed on $f$ to arrive at such induced maps in higher dimensions?
 A: Let's address pullbacks and pushforwards separately.
Pullback:
Whenever $\text{Cl}(X)$ agrees with the group of line bundles $\text{Pic}(X)$ (which happens for smooth varieties among others), you can simply pull back line bundles to get $f^* : \text{Pic}(Y) \to \text{Pic}(X)$. 
If you want to pull back divisors and not divisor classes, you need to impose certain conditions. One possible such condition (when $X$ is reduced) is that to pull back $D$ a divisor on $Y$, no component of $X$ can map into the support of $D$. More generally, to pull back a divisor $D$ on $Y$, no "associated subscheme" of $X$ (a subscheme defined by an associated prime of $\mathcal{O}_X$) should map into the support of $D$. 
Pushforward: 
Pushing forward divisors is typically not going to give you a divisor for higher-dimensional varieties. Take for example, $f : X \to \mathbb{P}^2$, the blowup of $\mathbb{P}^2$ at a point. The image of the exceptional divisor $E$ is a point and thus can't be defined as an element of the divisor class group. 
If you're willing to let things map to zero, then there's no issue defining pushforward whenever $f$ is a proper morphism. In fact, if $f$ is a proper morphism of schemes, you get an induced pushforward map of Chow groups. You also can get an induced pullback of Chow groups when $f$ is a flat morphism of constant relative dimension.
All this and more can be discovered in the early chapters of Bill Fulton's book "Intersection Theory". 
