The residue pairing of a line bundle with the dual canonical twist

Assume the Axiom of Choice as inherited from the residue suppliers (The Axiom of Choice). Let $C$ be a smooth proper geometrically integral curve over a perfect field $k$, and let $\mathcal{L}$ be an invertible ${\mathcal{O}}_C$-module with dual ${\mathcal{L}}^{-1}$ (Invertible sheaves). Write ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}^1$ for the canonical bundle (Canonical bundle and canonical divisors) and ${\mathcal{L}}_{\eta }\mathrm{/}\mathcal{L}$ for the sheaf of principal parts of $\mathcal{L}$ (Principal parts of an invertible sheaf on a curve). Tensoring with the dual gives the invertible sheaf ${\omega }_C\otimes {\mathcal{L}}^{-1}$ (Tensor product of sheaves of modules, The internal Hom sheaf of two module sheaves).

Local pairing. Let $c_p\in {\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$ be a local principal part of $\mathcal{L}$ at a closed point $p$, and let $s\in ({\omega }_C\otimes {\mathcal{L}}^{-1}{)}_p$ be a regular local section of the dual twist. Their product is a rational differential, defined modulo regular differentials at $p$: if $c_p$ is represented by ${\tilde{c}}_p\in {\mathcal{L}}_{\eta }$, then ${\tilde{c}}_{p}s$ is a rational differential. Replacing ${\tilde{c}}_p$ by another representative changes it by an element of ${\mathcal{L}}_p$, whose product with $s$ is in ${\omega }_{C,p}$ and is regular. Thus the local residue ${\mathrm{res}}_p(c_{p}s)$ of Residue of a rational differential at a separable closed point is independent of the representative of $c_p$. In a local trivialization with parameter $t$, this is the residue of the Laurent expansion of ${\tilde{c}}_{p}s$; only its $t^{-1}dt$ coefficient is used.

The pairing. Let $c=(c_p)$ be a finite-support family of local principal parts of $\mathcal{L}$; such families represent the classes of $H^1(C,\mathcal{L})$, namely $H^1(C,\mathcal{L})$ is the cokernel of the diagonal map ${\mathcal{L}}_{\eta }\to {⨁}_p{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$ (H^1 of a line bundle on a curve as principal parts modulo meromorphic and regular sections). Let $s$ be a global section of ${\omega }_C\otimes {\mathcal{L}}^{-1}$. Define $⟨c,s⟩:={\sum }_p{\mathrm{res}}_p(c_{p}s)\in k\mathrm{.}$ The sum is finite: $c_p=0$ for all but finitely many $p$ by finite support, and therefore ${\mathrm{res}}_p(c_{p}s)=0$ for all but finitely many $p$; a global section $s$ is regular at every closed point, so no further poles appear. The construction is $k$-linear in the family $c$ and in the section $s$, by the $k$-linearity of the local residue and of the tensor product.

This defines the residue pairing on pairs of representatives, $⟨,⟩\text{\hspace{0.17em}⁣}:(\text{finite-support families in }{⨁}_p{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p)\times H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})⟶k\mathrm{.}$ It is independent of the chosen representative of the principal-part family: a new representative differs by the principal parts of a global meromorphic section of $\mathcal{L}$ and by a family of local regular sections, and the independence is the descent statement proved in the next item of this development, which yields the induced $k$-bilinear pairing $H^1(C,\mathcal{L})\times H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})\to k\mathrm{.}$