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 $K_Y=f^*{\rm div}(dx)$ are canonical divisors of $X$ and $Y$, then $$ K_X = n\cdot K_Y + R. $$ For ease of reference I will call this the "strong" Hurwitz formula.

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).

This has led to me the following questions:

- Is it possible to prove this "strong" Hurwitz without using sheaves etc in the wildly ramified case?
- If so, are there any references that would help with this?
- Is there a different name for this "strong" Hurwitz formula.

I imagine/assume that an elementary proof would rely on computing the order of $dx$ at a point, but the best I am able to get from this is a lower bound on the order, not the precise value.

Previously asked at Stack Exchange (https://math.stackexchange.com/questions/174168/elementary-proof-of-the-hurwitz-formula/174189) but with no joy.