Normalization of the trace for Serre duality on a curve

Assume the Axiom of Choice (The Axiom of Choice), inherited from the fixed-trace, duality, cohomology and residue suppliers below. Let $C$ be a smooth proper geometrically integral curve over a perfect field $k$, and put ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}^1$ (Canonical bundle and canonical divisors). The trace $t_C$ is fixed by the projective Gysin construction in Dualizing line bundle and trace datum of a smooth projective variety and the published smooth-projective duality theorem (Serre duality for locally free sheaves on a smooth projective variety). It is not selected after a residue formula is chosen.

The functional $t_C^{\mathrm{r}\mathrm{e}\mathrm{s}}(\xi )={\sum }_{p\in C}{\mathrm{res}}_p({\xi }_p),\xi \in H^1(C,{\omega }_C),$ is well defined on principal-parts classes by the global residue theorem and The residue pairing is well defined on cohomology. The comparison proved in Serre duality for line bundles on a smooth proper curve, and the residue realization is $t_C=-t_C^{\mathrm{r}\mathrm{e}\mathrm{s}}$ over perfect fields. In outline, published duality applied to ${\mathcal{O}}_C$ and $H^0(C,{\mathcal{O}}_C)=k$ give ${\dim }_k{H}^1(C,{\omega }_C)=1$. At a closed point $p$ with $L=\kappa (p)$, choose $a\in L$ with ${\mathrm{Tr}}_{L\mathrm{/}k}(a)\mathrm{\ne }0$. After extension to an algebraic closure $K$, the finite separable algebra splits as $L{\otimes }_{k}K\cong {\prod }_{\sigma :L\hookrightarrow K}K$. On each resulting rational point, the point-last ordered Čech boundary of $u^{-1}du$ has raw Koszul cocycle $e_u\mapsto +du$. The fixed-Gysin trace-one point class has cocycle $e_u\mapsto -du$. The one-point sheaf-Ext spectral-sequence argument in the cited theorem proves this comparison in global ${\mathrm{Ext}}_{{\mathcal{O}}_C}^1(\kappa (x),{\omega }_C)$, not merely after localization. Thus the positive Yoneda boundary is the negative of the trace-one point class, with exactly one sign conversion. Trace base change then gives $t_C({\delta }_p(a{t}^{-1}dt))=-{\sum }_{\sigma }\sigma (a)=-{\mathrm{Tr}}_{L\mathrm{/}k}(a)$, while the coefficient-trace residue sum is $+{\mathrm{Tr}}_{L\mathrm{/}k}(a)$. This nonzero class spans $H^1(C,{\omega }_C)$, proving the asserted negative comparison without a free scalar choice.

The residue realization for general invertible $\mathcal{L}$ is also proved there: multiplication by a section of ${\omega }_C\otimes {\mathcal{L}}^{-1}$ forms a commutative diagram between the $\mathcal{L}(D)$ and ${\omega }_C(D)$ Cartier-twist exact sequences. Naturality identifies the lower connecting class with cup product, and the local residue formula evaluates it. Thus the fixed-trace pairing is the negative of the positive residue pairing. The coefficient-trace assertion here is scoped to perfect $k$, where every closed-point residue extension is separable; the fixed Gysin trace itself is defined over arbitrary fields.

The residue-field-valued coefficient residue ${\mathrm{Res}}_p$ satisfies ${\mathrm{Res}}_p(t^{-1}dt)=1\in \kappa (p)$. The $k$-valued residue used here is ${\mathrm{res}}_p={\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}\circ {\mathrm{Res}}_p$, so ${\mathrm{res}}_p(t^{-1}dt)={\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}(1)=[\kappa (p):k]\cdot 1_k$; it equals $1$ at a $k$-rational point, and may be zero when the characteristic divides the residue degree. Tate’s curve residue results supply the finite-extension trace formulas. The comparison of this local residue orientation with this library’s fixed projective Gysin orientation is the explicit Koszul and Čech calculation above; a computation on ${\mathbf{P}}^1$ alone would not establish the comparison on an arbitrary curve.