In Nakayama's book 2004, pg. 39, a formula is written:

Let $f:X \to Y$ be a generically finite and proper surjective morphism, $D$ be a Cartier divisor on $Y$. Then, $f_*f^*D=(\deg f) D$.

More precisely, we can compute in this way. Let $\sum_i D_i$ be a reduced divisor on $X$ of all prime divisors mapped onto $D$. Then, $f^*D= \sum_i r_iD_i$ where $r_i$ is the ramification index of $D_i$. Since $f_*D_i =(\deg f|_{D_i}) D$ by definition, we have $f_*f^*D= \sum_i r_i (\deg f|_{D_i}) D =(\deg f) D$.

Here is my question:

(1) why define $f_*D_i =(\deg f|_{D_i}) D$?

(2) why $\sum_i r_i(\deg f|_{D_i}) =\deg f$?

where can I find a detailed reference for these formulae? I try to search on Stacks but I cannot find a detailed answer.

Thank you!

  • 3
    $\begingroup$ Fulton, intersection theory. $\endgroup$
    – abx
    Feb 28, 2020 at 10:23


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