Intersection Theory flat/proper commutation I have a problem with proposition 1.7 from Fulton's book, "Intersection Theory", that states that if $f$ is a proper morphism between $k$-schemes, $X$ and $Y$, and if we make a flat base change $g: Y'\to Y$, then $g^\ast f_\ast=f'_\ast g'^\ast$, where g' and f' are the maps that complete the fiber square.
I really dont understand the proof, nor the reduction to the commutative algebra neither the commutative algebra per say. I'm really confused about this, so could somebody explain?
 A: I was working through this book recently, and I believe I can continue where Bananeen left off. Here's the computation (please notify me if I'm wrong).
Let $X_i'$ be the irreducible components of $X'$, and $Y_j'$ be the irreducible components of $Y'$. Note that the image of each $X_i'$ under $f'$ is some irreducible component $Y_j'$ of $Y'$. By definition, we have \begin{align}
[X'] = \sum_i{\ell_{\mathscr{O}_{X_i',X'}}(\mathscr{O}_{X_i',X'}) \cdot [X_i']}  = \sum_j\sum_{f(X_i')=Y_j'}{\ell_{\mathscr{O}_{X_i',X'}}(\mathscr{O}_{X_i',X'}) \cdot [X_i']}.
\end{align}
Therefore, \begin{align}
f'_\ast[X'] &= \sum_j{\Big(\sum_{f(X_i')=Y_j'}{\ell_{\mathscr{O}_{X_i',X'}}(\mathscr{O}_{X_i',X'}) \cdot [\kappa(\mathscr{O}_{X_i',X'}):\kappa(\mathscr{O}_{Y_j',Y'})]}\Big)\cdot [Y_j']} \\
&= \sum_j{\ell_{\mathscr{O}_{Y_j',Y'}}(\mathscr{O}_{Y_j',Y'} \otimes_K L) \cdot [Y_j']} \\
&= [L:K] \cdot \sum_j{\ell_{\mathscr{O}_{Y_j',Y'}}(\mathscr{O}_{Y_j',Y'}) \cdot [Y_j']} \\
&= d \cdot [Y'],
\end{align}
as desired. In the computation above, $\kappa$ denotes the residue field. As noted by Bananeen, Lemma A.1.3 (in the text) needs to be extended to the semi-local ring case, for the second equality above (note that $\mathscr{O}_{Y_j',Y'} \otimes_K L$ is a semi-local ring whose maximal ideals correspond to irreducible components $X_i'$ of $X'$ such that $f(X_i')=Y_j'$). The third equality follows, for example, from $\kappa(\mathscr{O}_{Y_j',Y'}) \otimes_K L \simeq \kappa(\mathscr{O}_{Y_j',Y'})^{[L:K]}$.
A: I agree that Fulton is too sketchy in chapter 1; StacksProject managed to simplify the exposition somewhat, that is true.
Now for the reduction in the book, contemplate the following diagram, where I take $W$ to be a subvariety in $X$:

Here is some explanation about what is going on in there:

*

*$f(W)^{sch}$ denotes the scheme-theoretic image;

*rear facet is cartesian by assumption;

*upper and lower facets are cartesian by construction;

*left-most arrow exists by the universal property of $g'^{-1}(f(W)^{sch})$ (you need to play the diagram a bit);

*front-right arrow is actually proper, via an application of 10.3.4.(e) of Vakil's notes ($W \to X \to Y$ is proper, $f(W)^{sch} \to Y$ is separated by 10.1.B);

*the front facet is in fact cartesian by 2,3 and the commutativity of the diagram; thus front-left arrow is proper.

This completes the reduction in the first line of Fulton's explanation, so we assume $X,Y$ are varieties and map between them is proper surjective.
As far as commutative algebra goes, notice that the assumption that horizontal maps are flat of some relative dimension $n$ implies that, if $V_i^{'}$ is a connected component of $Y'$, then $g(V_i^{'})$ is an irreducible component of $Y$ (see proof of lemma 1.7.1), thus $g(V_i^{'})=Y$ (remember that $Y$ is irreducible).
Thus we obtain maps of local rings $K \to \mathcal{O}_{V_i^{'}, Y'}$. Denote the latter for a fixed $i$ by $A$, so that $A$ now becomes Artinian local ring. Complete the cocartisian square:

Now, as in the proof of proposition 1.4 one shows that $B$ is a semi-local ring, whose maximal ideals correspond to irreducible components of $X'$ dominating $V_i^{'}$.
Now in principle we have everything to show the desired equality; the only recommendation is to use the generalization of Lemma A.1.3 to semi-local rings in the form of this tag.
[The only place I feel shaky is showing that degrees of field extensions we get for components of $X' \to Y'$ coincide with $d$, which is the degree of extension $[R(X):R(Y)]$. Because of that, I'll leave the final piece of proof for the time being. Any ideas would be appreciated.]
