Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to see that $2c_1(\mathscr O_X)=-f_*R_f$ where $R_f$ is the raminification divisor of $f,$ and I got a great answer explaining just that. A comment of Yusuf Mustopa caught my interest though.

The trace pairing gives rise to an injection $f_*\mathscr O_X\to \mathscr{H}om_{\mathscr O_Y}(f_*\mathscr O_X,\mathscr O_Y)$ (see Jason Starr's answer to the previous question for details). Via the Thom-Porteous formula, my previous question boils down to showing that the degeneracy divisor of this morphism of vector bundles is linearly equivalent to the divisor $f_*R_f.$

*Can that be seen explicitly?*

By that I mainly mean: not by translating it to statements about norms, discriminants and differents as implicitly done in Jason Starr's answer mentioned above.

Thanks in advance!

equalas effective Cartier divisors. As Will says, this stronger claim is a local statement that can even be checked after base change; after 'etale base change, you can arrange that the target is a DVR and the domain is a product of DVRs. In characteristic $0$, you only need to consider $x\mapsto x^n$, just as Will suggests. However, in char $p>0$, there are many more cases (moduli of non-tame covers). Best advice: read Serre. $\endgroup$ – Jason Starr Mar 5 '14 at 10:53