Ramification formula for orbifolds It's well known for smooth curves that if $\pi:X\to Y$ is a finite map, $K_X=\pi^*K_Y+Ram(\pi)$, this is just the Riemann-Hurwitz formula at the level of line bundles.  I've been told that this formula is true for a finite morphism of smooth varieties over $\mathbb{C}$, at least.
First, is the above true?
Second, how must the formula be corrected if $X$ and $Y$ are both smooth orbifolds/DM-stacks?
In particular, what if I want to use that $X$ and $Y$ are of the forms $\tilde{X}/G$ and $\tilde{Y}/H$ for groups $G,H$, and express the formula in terms of objects on $\tilde{X},\tilde{Y}$?
 A: In order for $K_X$ to exist you need some restriction on the singularities of $X$. See this MO answer for some thoughts on that. The most usual assumption to make is that $X$ is normal which implies that it is non-singular in codimension $1$.
In order to pull-back $K_X$ and still get a divisor, you need something more, the usual assumption is that it should be $\mathbb Q$-Cartier. See this MO answer for more on that.
If you have that $X$ and $Y$ are non-singular in codimension $1$, then you can forget about the second condition and just consider the formula on the non-singular part. (Strictly speaking you would want to restrict to the complement of the image of the singular part of $X$ on $Y$ so you'd still have a finite map).
So, let's say that both $X$ and $Y$ are smooth and $\pi$ is finite. Then localizing at height $1$ primes (or codimension $1$ points) reduces the statement to curves. Albeit not over the original field, but it's still OK. 
I think this should cover two of your questions.
