Functoriality of the residue pairing under line-bundle maps and connecting homomorphisms

The multiplication map ${\mu }_t$ is well defined: if the finite-support families $(c_p)$ and $(c_p^{\mathrm{\prime }})$ both represent the class $c\in H^1(C,\mathcal{L})$, then $(c_p-c_p^{\mathrm{\prime }})$ is a sum of the principal parts of a global meromorphic section $\theta \in {\mathcal{L}}_{\eta }$ and of a family with $c_p\in {\mathcal{L}}_p$ for all $p$, by [F1]; multiplying by the global section $t$ gives the principal parts of $\theta \otimes t\in (\mathcal{L}\otimes \mathcal{M}{)}_{\eta }$ together with a family with $t{c}_p\in (\mathcal{L}\otimes \mathcal{M}{)}_p$, so $(t{c}_p)$ is again a finite-support family representing a class of $H^1(C,\mathcal{L}\otimes \mathcal{M})$ independent of the representative; the Axiom of Choice is used only as inherited from the residue suppliers of [F1], and ${\mu }_t$ is $k$-linear because multiplication by $t$ is $k$-linear.

Multiplicativity on representatives: for a finite-support family $(c_p)$ representing $c$ and the global section $s$ of ${\omega }_C\otimes {\mathcal{L}}^{-1}$ with $s\otimes t^{-1}$ regular, the family $(c_p\otimes t)$ represents ${\mu }_t(c)$ and $(c_p\otimes t)\cdot (s\otimes t^{-1})=c_p\cdot s\otimes (t\otimes t^{-1})$ inside $(\mathcal{L}\otimes \mathcal{M}{)}_{\eta }\otimes ({\omega }_C\otimes (\mathcal{L}\otimes \mathcal{M}{)}^{-1}{)}_{\eta }$; under the canonical identification of [F2] with ${\omega }_C\otimes (\mathcal{M}\otimes {\mathcal{M}}^{-1}{)}_{\eta }$ and then with ${\omega }_{C,\eta }$, the factor $t\otimes t^{-1}$ becomes $1$, so $c_p\otimes t\cdot s\otimes t^{-1}$ and $c_{p}s$ have the same image in ${\omega }_{C,\eta }={\mathrm{\Omega }}_{K\mathrm{/}k}^1$, hence the same residue at every closed point $p$.

The twist sequence and its connecting map: by [F3] the sequence $0\to \mathcal{L}\to {\mathcal{L}}^{\mathrm{\prime }}\to i_{\ast }({\mathcal{L}}^{\mathrm{\prime }}{\mathrm{\mid }}_D)\to 0$ is exact, its quotient is a finite-length skyscraper with global sections ${⨁}_{p\in \mathrm{supp}D}{\mathcal{L}}_p^{\mathrm{\prime }}\mathrm{/}{\mathcal{L}}_p$, and the connecting homomorphism $\delta \text{\hspace{0.17em}⁣}:H^0(C,i_{\ast }({\mathcal{L}}^{\mathrm{\prime }}{\mathrm{\mid }}_D))\to H^1(C,\mathcal{L})$ is computed by local lifting; the support $\mathrm{supp}D$ is finite because it is a closed subset of the one-dimensional Noetherian curve $C$ different from $C$.

Therefore $⟨{\mu }_t(c),s\otimes t^{-1}{⟩}_{\mathcal{L}\otimes \mathcal{M}}={\sum }_p{\mathrm{res}}_p(c_p\otimes t\cdot s\otimes t^{-1})={\sum }_p{\mathrm{res}}_p(c_{p}s)=⟨c,s{⟩}_{\mathcal{L}}$, since the sums are finite by [F1] and the residues agree term by term by step 1.2; this is the asserted commutativity of the diagram.

Description of $\delta (g)$ by lifts: put $Q:=i_{\ast }({\mathcal{L}}^{\mathrm{\prime }}{\mathrm{\mid }}_D)$ and $\mathcal{P}(\mathcal{L}):={\mathcal{L}}_{\eta }\mathrm{/}\mathcal{L}$. The inclusion ${\mathcal{L}}^{\mathrm{\prime }}\hookrightarrow {\mathcal{L}}_{\eta }$ and quotient maps give a commutative diagram of short exact sequences with identity on $\mathcal{L}$ and the induced map $Q\hookrightarrow \mathcal{P}(\mathcal{L})$ on the right. Naturality of the connecting homomorphisms in the long exact cohomology sequences [F3] sends $\delta (g)$ to the boundary class of the image of $g$ in $H^0(C,\mathcal{P}(\mathcal{L}))$. By the principal-parts presentation [F1], that image is the finite-support family whose component at each $p\in \mathrm{supp}D$ is ${\tilde{g}}_p+{\mathcal{L}}_p$ and whose other components are zero, and its boundary class is represented by precisely this family in $H^1(C,\mathcal{L})$. Thus the chosen local lifts represent $\delta (g)$ under [F1]; changing a lift changes it by a regular element and does not change its principal-part class.

If the global section $t$ is nowhere vanishing, then $t^{-1}$ is a global section of ${\mathcal{M}}^{-1}$, so $s\otimes t^{-1}$ is a global section of ${\omega }_C\otimes {\mathcal{L}}^{-1}\otimes {\mathcal{M}}^{-1}={\omega }_C\otimes (\mathcal{L}\otimes \mathcal{M}{)}^{-1}$ for every global section $s$ of ${\omega }_C\otimes {\mathcal{L}}^{-1}$; moreover ${\mu }_t$ and ${\mu }_{t^{-1}}$ are mutually inverse because $t\otimes t^{-1}=1$ under the duality pairing and ${\mu }_t{\mu }_{t^{-1}}={\mu }_{t\otimes t^{-1}}=\mathrm{i}\mathrm{d}$ on classes, so ${\mu }_t$ is an isomorphism identifying the pairings by step 2.1.

Evaluation of the pairing: for $s\in H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})$, step 2.2 and [F1] give $⟨\delta (g),s{⟩}_{\mathcal{L}}={\sum }_{p\in \mathrm{supp}D}{\mathrm{res}}_p({\tilde{g}}_{p}s)$, the sum being finite by step 1.3.

Independence of the lifts: if ${\tilde{g}}_p$ and ${\tilde{g}}_p^{\mathrm{\prime }}$ both lift $g_p$, then ${\tilde{g}}_p-{\tilde{g}}_p^{\mathrm{\prime }}\in {\mathcal{L}}_p$, so $({\tilde{g}}_p-{\tilde{g}}_p^{\mathrm{\prime }})s\in {\mathcal{L}}_p\cdot ({\omega }_C\otimes {\mathcal{L}}^{-1}{)}_p={\omega }_{C,p}$ is a regular differential and has residue zero. Thus the sum in step 3.2 is independent of the lifts and defines the map ${\rho }_D$ in the statement.

(Adjunction and the annihilator of the local restriction.) Step 3.2 identifies $⟨\delta (g),s{⟩}_{\mathcal{L}}$ with ${\rho }_D(s)(g)$ for every $g$ and $s$, so $\delta$ is adjoint to ${\rho }_D$. To compute its kernel, fix $p\in \mathrm{supp}D$, choose a uniformizer $t$, and trivialize $\mathcal{L}$ by $\mathrm{\ell }$ and ${\omega }_C$ by $\mathrm{d}t$. If the multiplicity of $D$ at $p$ is $e$, then ${\mathcal{L}}_p={\mathcal{O}}_{C,p}\mathrm{\ell }$ and ${\mathcal{L}}_p^{\mathrm{\prime }}=t^{-e}{\mathcal{O}}_{C,p}\mathrm{\ell }$ up to a unit, so ${\mathcal{L}}_p^{\mathrm{\prime }}\mathrm{/}{\mathcal{L}}_p$ is spanned over $\kappa (p)$ by the classes $t^{-j-1}\mathrm{\ell }$ for $0\le j<e$. Write the regular section $s$ as $u(t){\mathrm{\ell }}^{-1}\mathrm{d}t$ in the completion, with $u(t)={\sum }_{m\ge 0}{u}_m{t}^m$. For each $j$ and $a\in \kappa (p)$, lift $a$ to $\tilde{a}\in {\mathcal{O}}_{C,p}$ and test against the class of $t^{-j-1}\tilde{a}\mathrm{\ell }$. Inductively, if $u_0=\cdots =u_{j-1}=0$, the coefficient of $t^{-1}\mathrm{d}t$ in this product is $a{u}_j$: terms of positive order in $\tilde{a}$ multiply only coefficients $u_m$ with $m<j$. By nondegeneracy of the trace form in [F7], if $u_j\mathrm{\ne }0$ one can choose $a$ with ${\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}(a{u}_j)\mathrm{\ne }0$. Thus all residue tests against ${\mathcal{L}}_p^{\mathrm{\prime }}\mathrm{/}{\mathcal{L}}_p$ vanish exactly when $u_0=\cdots =u_{e-1}=0$, that is, when $s$ lies in $({\omega }_C\otimes {\mathcal{L}}^{\mathrm{\prime }-1}{)}_p=({\omega }_C\otimes {\mathcal{L}}^{-1})(-D{)}_p$. The quotient sheaf is a direct sum over the support, so these tests at each $p$ show that $\ker {\rho }_D$ is the image of $H^0(C,{\omega }_C\otimes {\mathcal{L}}^{\mathrm{\prime }-1})$.

Combining steps 2.1, 3.2, 4.1 and 5.1 proves the connecting-map formula, lift independence, adjunction and kernel statement in (2); steps 1.2 and 2.1 prove (1), including the nowhere-vanishing case.