Let $f:X \rightarrow Y$ be a finite, separable morphism of curves (curve: integral scheme, of dimension 1, proper over an algebraically closed field with all local rings regular). Let $R$ be the ramification divisor on $X$.
Question 1: What is exactly $f_*(R)$? If $R = \sum_{P \in X} r_P \cdot P$ ($r_P$ is the ramification index), then i suppose that $f_*(R) = \sum_{P \in X} r_P \cdot f(P)$. Is that correct?
Now let $B$ be a boundary of $Supp(f_*(R))$, i.e. a divisor on $K_Y$ with nonzero coefficients equal to $1$ on a superset of the support of $f_*(R)$. We know that there is a homomorphism $f^* : Div(Y) \rightarrow Div(X)$ (see Hartshorne bottom of page 137) and so we can talk about $f^* B$.
Question 2: What is $f^{-1} B$?
Motivation: i saw this notation in an exercise where i am asked to prove a Hurwitz-like formula of the form $f^*(K_Y+B)=K_X+f^{-1}B$, where $K_X,K_Y$ are the canonical divisors.
Best Answer
(1) Yes, this is a standard definition. You should have read it before.
(2) This is not a canonical notation. Can you check your exercise ?