In Fulton's Intersection Theory, Example 1.9.5. Let $f:X'\to X$ be a finite, birational morphism of $n$-dimensional varieties. For each codimension one subvariety $V$ of $X$, let $d(V)$ be the greatest common divisor of the degrees of all field extensions $R(W)$ of $R(V)$, for all subvarieties $W$ of $X'$ such that $f(W)=V$. There is an exact sequence
I wonder how to obtain this exact sequence. I can see that the image of the pushforward map $f_*: A_{n-1}X'\to A_{n-1} X$ is generated by $d(V)[V]$.
The first map is obviously the proper pushforward. For any codimension 1 subvariety $ W \subset X' $ mapping to a codimension 1 subvariety $ V \subset X $, the relation $ f_* [W] = [K(W):K(V)] . [V] $ shows that the coefficient of $ [V] $ in any codimension 1 cycle of $ X $ coming from $ f_* $ is in the ideal generated by all $ [K(W):K(V)] $, that is by $ d(V) $.
The second map sends a codimension 1 cycle $ \sum n_V [V] $ of $ X $ to the collection $ n_V \pmod{d(V)} $ in the direct sum (because a cycle has only finitely many non-zero coefficients. It only remains to show that cycles of the form $ \operatorname{div}(r) $ go to zero, for a non-zero rational function $ r \in K(X) $. But $ f $ is birational so $ K(X) \cong K(X') $ and the norm map is trivial. Let $ r' $ be the image of $ r $ in $ X' $ so that $ f_* \operatorname{div}(r') = \operatorname{div}(N(r)) = \operatorname{div}(r) $ and the first paragraph now shows that $ \operatorname {div}(r) $ is sent to zero under the second map.
Somehow I've never used the fact that $ f $ is finite. I think only proper+birational suffices.