Degree of a Cartier Divisor under pullback This is question 7.2.3 in Liu's book Algebraic Geometry and Arithmetic Curves and I have been trying with this for some time now.
Let $f:X \rightarrow Y$ be a morphism of Noetherian schemes, and suppose that X and Y are integral and that f is finite surjective. We will let $Div(Y)$ resp. $Div(X)$ stand for Cartier Divisors on Y resp. X. 
Associated to a Cartier divisor $D \in Div(Y)$ we have a notion of multiplicity at a point $y \in Y$, defined as $$\text{mult}_y D = \text{length } \mathcal{O}_{Y,y}/D_y \mathcal{O}_{Y,y}.$$
I am trying to show that for a Cartier Divisor $D \in Div(Y)$, we have that $$\text{mult}_x f^\ast(D) = \text{length } \mathcal{O}_{X,x}/(\mathcal{m}_y\mathcal{O}_{X,x}) \cdot \text{mult}_y(D)$$
where $m_y$ is the maximal ideal of $\mathcal{O}_{Y,y}$. 
Now, it is quite easy to do this if f is a flat morphism, since we can simply use 50.13 of this document http://stacks.math.columbia.edu/download/algebra.pdf (Stacks, Algebra). But I don't see how to do it in the general case, and I have been trying for some time, so any hint would be nice. 
Thank you!
 A: I just e-mailed Qing Liu about this exercise and got a nice reply (as always), and I will share it in case other people try this exercise and get stuck. Please excuse that I refer to numbers in the book
"7.2.3 a) is incorrect for the case f finite surjective, see EGA IV 5.6.11 . Maybe it can be corrected by assuming further that Y is normal.
In b) one should keep the hypothesis that Y is normal. "
I think I know how to do this in this case and will post a solution then, by editing this answer.
Update:
OK - here is how I think one can do it in the case Y is normal. Let us note that the property is local on Y, so that we can assume Y to be a normal integral, local noetherian ring of dimension 1. As such, Y is a Dedekind scheme. Now, we know that for a non-constant morphism $f: X \rightarrow Y$ with X integral and Y Dedekind , f is flat (see Liu, Corollary 4.3.10) . Hence, it reduces to the flat case and this is an easy exercise in commutative algebra.
And I forgot to add, if I am wrong, please tell me!
