Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective varieties over a field $k$ of characteristic zero, where $\dim X=\dim Y$. Then $f$ is flat. Hence $f_\ast \mathcal{O}_X$ is a coherent locally free sheaf on $Y$.

Now, my question is based on the following example. (For simplicity, take $k=\overline{k}$.)

**Example**. Suppose that $\dim X =\dim Y = 1$ (i.e., curves). Apply Grothendieck-Riemann-Roch to $f$ and $\mathcal{O}_X$. In degree 0, we get the fact that $\textrm{deg} \ f = \textrm{rk} \ f$. In degree 1 we get a "Hurwitz theorem". In fact, with little effort the formula reads $$2c_1(f_\ast \mathcal{O}_X) =\deg f \cdot K_Y - f_\ast(K_X) = f_\ast(-R), $$ where $R$ is the ramification divisor on $X$.

Now for my two questions that are based on this formula.

**Q1**. The divisor $R$ is not called the ramification divisor for nothing. Its support is the set of ramification points and the multiplicity of $R$ at a point $P$ is precisely $e_P-1$. So in my opinion, it "measures" the ramification. What about $c_1(f_\ast \mathcal{O}_X) = c_1(\det f_\ast \mathcal{O}_X)$? How does he "measure" the ramification? (I'm probably missing something really elementary here.)

**Q2**. In higher-dimensions, if I understand correctly, one should get a "higher-dimensional" Hurwitz formula: $$2c_1(f_\ast \mathcal{O}_X) =f_\ast(\textrm{td}(X/Y)).$$ I doubt that this "measures" all the ramification. And, to be frank, I don't really know what it "measures". Can anyone provide some insight?