# Show that $f_\ast f^\ast[V]=d[V]$, and find some examples.

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 $$f_\ast f^\ast[V]=\sum_{W\subset f^{-1}(V)}n_W\deg(W/f(W))[f(W)],$$ 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 $$f_\ast f^\ast[V]=\sum_{W\subset f^{-1}(V)}n_W\deg(W/V)[V].$$ 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?

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A nice example to work out is to consider a ramified cover of Riemann surfaces (curves over $\mathbb{C}$), $\pi: C_1\to C_2$. The reason to pick something ramified is to see what is going on in the formula. For all the unramified points $p\in C_2$, when you pull back you get $d$ distinct points so everything is fine. At ramified points the pullback gives you less than $d$ distinct points set-theoretically, but because you remember more information scheme theoretically the points that "came together" will have multiplicity and hence give the right formula again. – Matt Sep 29 '12 at 16:48
OK @Matt, I see. So then, if $\pi$ is unramified then I can take any $y\in C_2$ and get $d=\sum_{x\in \pi^{-1}y}[k(x):k(y)]=|\pi^{-1}(y)|$. And if $\pi$ ramifies (necessarily at a finite set $S\subset C_2$) I get, for all $y\in S$, $d=\sum_{x\in \pi^{-1}y}e_{x/y}$, right? Question: when we say $\pi$ ramifies at $y\in C_2$, does it mean that for $\textbf{every}$ $x\in\pi^{-1}(y)$ one has $e_{x/y}>1$? Or may this hold just for some $x$? – Brenin Sep 30 '12 at 9:21
Just fir some $x$'s : take $C_1=C_2=\mathbb C$ and $\pi(z)=z(z-1)^2$. Then above the critical value $y=0$ the point $x=1$ is a ramification point but $x=0$ is not. – Georges Elencwajg Sep 30 '12 at 16:54
@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 $d=\sum e_{x/y}$. – Matt Sep 30 '12 at 21:16

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 $B=A\otimes_{O_Y(U)} O_X(f^{-1}(U))$.
Thank you, @QiL! I see that $B\cong A^d$, but: $\textbf{(1)}$ I don't see why a maximal ideal $\mathfrak n\subset B$ should correspond to an irreducible component $W$ of $f^{-1}(V)$. $\textbf{(2)}$ Let us suppose that $\mathfrak n\leftrightarrow W$ (so that the local ring of $W$ in $f^{-1}(V)$ is $B_{\mathfrak n}$). The multiplicity $n_W$ is defined as $\ell_{B_{\mathfrak n}}(B_{\mathfrak n})$, but the corresponding term in the sum is $\ell_{B_{\mathfrak n}}(B_{\mathfrak n}/\mathfrak m_AB_{\mathfrak n})=:e_{B_{\mathfrak n}/A}$. Is there any problem here? That is, are these numbers the same? – Brenin Oct 1 '12 at 19:48
@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 $B_{\mathfrak n}/(\mathfrak m_A)$. – user18119 Oct 1 '12 at 22:36