I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective non-singular curves, with genera $g_X$ and $g_Y$, then $ 2(g_X-1) = 2n(g_Y-1) + \deg(R), $ where $R$ is the ramification divisor of $f$. The proof of this was given to me as an exercise when I started my PhD, and I am very happy with it.
However, in some other work that I was doing it appeared that one could strengthen this to say rather that if $K_X={\rm div}(f^*(dx))$ and $K_Y={\rm div}(dx)$ are canonical divisors of $X$ and $Y$, then $ K_X = n\cdot K_Y + R. $ I have found this alluded to in a number of places, and even stated in Algebraic Curves Over Finite Fields by Carlos Moreno. However, this was without proof, and every idea of a proof that I have seen is in sheaf theoretic language. I am slowly getting through sheaves and schemes, but I am currently trying to prove this in an elementary manner (fiddling around with orders of $dx$ etc.), in the wildly ramified case (the tamely ramified case is fine).
I would like to know if
- It is possible to prove this without using sheaves etc in the wildly ramified case
- If so, are there any references that would help with this.
edit: Also, is there a different name for the "more specific" Hurwitz formula?