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I'm trying to understand Example 1.7.4 in Fulton's "Intersection Theory", asserting that if a morphism of schemes (of finite type over a field) $f:X\to Y$ is flat and finite of degree $d$, then for every subvariety $V\subset Y$ one has $f_\ast f^\ast[V]=d[V]$. My calculation leads to the equality \begin{equation} f_\ast f^\ast[V]=\sum_{W\subset f^{-1}(V)}n_W\deg(W/f(W))[f(W)], \end{equation} where the sum is on the irreducible components $W$ of $f^{-1}(V)$ and $n_W=\ell(\mathscr O_{f^{-1}(V),W})$. Because $f^{-1}(V)\to V$ is flat and $V$ is irreducible, every $W$ dominates $V$. But as $f$ is finite (hence closed) we get $f(W)=V$ for all $W$, so we can write \begin{equation} f_\ast f^\ast[V]=\sum_{W\subset f^{-1}(V)}n_W\deg(W/V)[V]. \end{equation} Now, we are left to show that $\sum_Wn_W\deg(W/V)=d$. I know that if I have a local ring $A$ and an $A$-algebra $B\cong A^d$, then $d=\sum_{\mathfrak n\in\textrm{Spm}\,B}[B_\mathfrak n/\mathfrak nB_\mathfrak n:A/\mathfrak m_A]\ell_{B_\mathfrak n}(B_\mathfrak n/\mathfrak m_AB_\mathfrak n)$. Hence, to conclude I'd like to take $A=\mathscr O_{Y,V}$, but $\textbf{what}$ $B$ $\textbf{do I have to choose?}$ I'd like to interpret $n_W$ as the length appearing in the sum (so I need a correspondence $\mathfrak n\leftrightarrow W$) and to recover $\deg(W/V):=[R(W):R(V)]$ as the degree $[B_\mathfrak n/\mathfrak nB_\mathfrak n:A/\mathfrak m_A]$.

Also, can you show me an $\textbf{example}$ of this result?

Thank you in advance.

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    @atricolf My terminology wasn't quite standard. I should have called $p\in C_2$ for which some $\pi^{-1}(p)$ are ramified a branch point. The pullback $\pi^*(p)=\sum_{x\in \pi^{-1}p} e_{x/p}x$ and only some of the $x$ will necessarily have e_{x/y}>1. It is true that $d=\sum e_{x/y}$.2012-09-30

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Let $\xi$ be the generic point of $V$. Then $O_{Y,V}=O_{Y,\xi}$. Let $U$ be an affine open neighborhood of $\xi$. Then take $B=A\otimes_{O_Y(U)} O_X(f^{-1}(U))$.

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    @atricolf: (1) The maximal ideals in $B$ are over the maximal ideal $A$ by the finiteness of $B$ over $A$. So the corresponding points are over the generic points of $V$. As $f$ is flat, $f^{-1}(V)\to V$ is flat, so the irreducible components of $f^{-1}(V)$ are the Zariski closure of the points of the generic fiber. (2) non the multiplicity is not the length of $B_{\mathfrak n}$ (which is not of finite length!), but the length of $B_{\mathfrak n}/(\mathfrak m_A)$.2012-10-01