Let $f:X\to Y$ be a finite morphism of curves. Let $L$ be a line bundle on $X$. Why is $f_\ast L$ a line bundle and is the degree of $f_\ast L$ equal to $\deg f$ or $\deg f+ \deg L$?
Here is my strategy and some related questions.
Firstly, since $f$ is proper, we have that $f_\ast L$ is coherent on $Y$. I want to see that it is locally free. I have read that this follows from the flatness, but this is too abstract for me.
Can't we do something more explicit? The problem is local on $Y$. Suppose that $V_1,\ldots,V_r$ are trivializing opens for $L$ on $X$. Then, how does one trivialize $f_\ast L$. Problems might occur at the branch point, but therefore I ask: if I show that $(f_\ast L)_y$ is free for all non-branch points $y$ of $f:X\to Y$, does it follow that $f_\ast L$ is locally free?
I don't want to use Grothendieck-Riemann-Roch or anything similar.