Trace-dual module of a finite curve map (${\pi }_{\ast }{\Omega }_C^1\cong \mathcal{H}om({\pi }_{\ast }{\mathcal{O}}_C,{\Omega }_D^1)$)

For a finite generically separable morphism between smooth projective curves, the field trace on rational differentials identifies the differential sheaf with the trace-dual module. Locally, for $A=k[[s]]\subset B=k[[u]]$ with $s=f(u)$, the trace-dual fractional ideal is $f^{\prime }(u{)}^{-1}B$. Multiplying it by $ds=f^{\prime }(u)du$ gives $Bdu$. For a monogenic finite extension with monic polynomial $P$, the identity $\mathrm{Tr}(g(u)/P^{\prime }(u))=[T^{e-1}](g(T)\mathrm{mod}P(T))$ proves the dual-module formula: the top-coefficient pairing on the basis $1,u,\dots ,u^{e-1}$ has invertible antitriangular matrix. Tensoring locally with an invertible sheaf yields ${\pi }_{\ast }({\mathcal{M}}^{\vee }\otimes {\Omega }_C^1)\cong \mathcal{H}om({\pi }_{\ast }\mathcal{M},{\Omega }_D^1)$.