Curve duality and its residue normalization in dimension one

Under AC, for a projective pure CM curve $C\mathrm{/}k$, the theorem gives $H^1(C,F{)}^{\vee }={\mathrm{Hom}}_C(F,{\omega }_C)$ and $H^0(C,F{)}^{\vee }={\mathrm{Ext}}_C^1(F,{\omega }_C)$ for every coherent $F$. In particular it includes singular curves. The second formula uses global Ext; replacing it by $H^1(C,F^{\vee }\otimes {\omega }_C)$ requires $F$ to be locally free. When $C$ is smooth and $E$ is a vector bundle, the preceding smooth specialization identifies ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}^1$ and the first pairing with $H^1(C,E)\times H^0(C,E^{\vee }\otimes {\mathrm{\Omega }}_{C\mathrm{/}k}^1)\to k$.

The normalization agrees with curve residue duality, including its sign. At a rational smooth point with parameter $z$, the boundary class of the principal part $dz\mathrm{/}z$ is the point Gysin class; in the signed derived/Koszul convention its dual top cochain is $e_z\mapsto -dz$. Its normalized trace is $1$ by Rational-point Koszul residue normalization for a smooth projective embedding, which also proves independence of the parameter and compatibility with field extension. This is the local residue convention $\mathrm{res}(dz\mathrm{/}z)=1$. After an algebraic closure of $k$, each smooth connected component has a rational point and its $H^1({\omega }_C)$ is one-dimensional, by the existing smooth theorem applied to ${\mathcal{O}}_C$. Thus these point normalizations determine the trace on every component. Compatibility with field extension descends the equality to $k$. Multiplication and evaluation with a bundle section commute with this point-class construction, so the specialization above uses the same residue-normalized trace as the curve pairing. The remark compares the dimension-one pairing and its normalization; it does not assign ordinary rational differentials as the dualizing sheaf of a singular curve.