# A formula on a generically finite morphism

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!

• Fulton, intersection theory.
– abx
Feb 28, 2020 at 10:23