= Trace-dual module of a finite curve map
{title2=$\pi_*\Omega^1_C\cong\mathcal H om(\pi_*\mathcal O_C,\Omega^1_D)$}
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 $f'(u)^{-1}B$. Multiplying it by $ds=f'(u)du$ gives $Bdu$. For a monogenic finite extension with monic polynomial $P$, the identity $\operatorname{Tr}(g(u)/P'(u))=[T^{e-1}](g(T)\bmod P(T))$ proves the dual-module formula: the top-coefficient pairing on the basis $1,u,\ldots,u^{e-1}$ has invertible antitriangular matrix. Tensoring locally with an <invertible sheaf> yields $\pi_*(\mathcal M^\vee\otimes\Omega^1_C)\cong\mathcal H om(\pi_*\mathcal M,\Omega^1_D)$.
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