A nonzero global dual section detects a cohomology class

Given: a perfect field $k$; a smooth proper geometrically integral curve $C$ over $k$, so that $C$ is a nonempty irreducible finite-type $k$-scheme of chain dimension $1$ (in the sense of Chain dimension and the empty-space convention) with generic point $\eta$, every closed point of $C$ has a finite residue field, and every nonempty open subset of $C$ contains $\eta$; an invertible ${\mathcal{O}}_C$-module $\mathcal{L}$; and a global section $s\in H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})$.

The sheaf ${\mathcal{L}}_{\eta }$ of meromorphic sections of $\mathcal{L}$ is the constant sheaf with value the one-dimensional $K$-vector space ${\mathcal{L}}_{\eta }$, $K=k(C)$; the stalk of the sheaf of principal parts $\mathcal{P}(\mathcal{L})={\mathcal{L}}_{\eta }\mathrm{/}\mathcal{L}$ at a closed point $p$ is ${\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$, and $H^0(C,\mathcal{P}(\mathcal{L}))={⨁}_p{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$ is the space of finite-support families of local principal parts. For an invertible sheaf on the integral curve, its map into the generic fibre is locally $A\hookrightarrow K$ in a frame and is injective. Thus a nonzero global section of ${\omega }_C\otimes {\mathcal{L}}^{-1}$ has a nonzero generic germ and a nonzero germ at every closed point. The principal-parts descriptions are supplied by Principal parts of an invertible sheaf on a curve (Invertible sheaves).

The cohomology space $H^1(C,\mathcal{L})$ is the cokernel of the diagonal map ${\mathcal{L}}_{\eta }\to {⨁}_p{\mathcal{L}}_{\eta }\mathrm{/}{\mathcal{L}}_p$; for a finite-support family $c=(c_p)$ and a global section $s\in H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})$ the sum $⟨c,s⟩={\sum }_p{\mathrm{res}}_p(c_{p}s)$ is finite, depends only on the class of $c$ in $H^1(C,\mathcal{L})$, and the induced residue pairing $H^1(C,\mathcal{L})\times H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})\to k$ is $k$-bilinear (H^1 of a line bundle on a curve as principal parts modulo meromorphic and regular sections, The residue pairing of a line bundle with the dual canonical twist, The residue pairing is well defined on cohomology).

At a closed point $p$ with $\kappa (p)$ finite separable over $k$, a rational differential written as $\omega =a\mathrm{d}t$ with $t$ a uniformizer and $a\in K\subseteq \kappa (p)((t))$ has residue ${\mathrm{res}}_p(\omega )={\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}([t^{-1}]a)$, the field trace of the coefficient of $t^{-1}$ in the Laurent expansion (Residue of a rational differential at a separable closed point).

The local ring ${\mathcal{O}}_{C,p}$ is a discrete valuation ring with uniformizer $t$, maximal ideal $(t)$, every nonzero element of it is a unit times a power of $t$, and the residue map ${\mathcal{O}}_{C,p}\to \kappa (p)$ is surjective with kernel $(t)$ (Local rings at closed points of smooth curves are discrete valuation rings). For every nonzero rational function $x\in K$, Every nonzero fraction is a unit times a power of a uniformiser gives the unique expression $x=w{t}^m$ with $w\in {\mathcal{O}}_{C,p}^{\times }$ and $m={\mathrm{ord}}_p(x)$; in particular, when $x$ is regular and nonzero, $m\ge 0$ and $t^{-m}x$ is a unit.

The canonical bundle ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}^1$ is invertible. At each closed point $p$, the perfect-field hypothesis makes $\kappa (p)\mathrm{/}k$ finite separable, so $\mathrm{d}t$ is a basis of ${\omega }_{C,p}$ for a uniformizer $t$; every rational differential has the form $g\mathrm{d}t$ for a unique $g\in K$. The local basis assertion is supplied by A uniformizer differential generates the module of differentials, and invertibility by Canonical bundle and canonical divisors.

If $k$ is perfect then every algebraic extension of $k$ is separable, so a finite residue field extension $\kappa (p)\mathrm{/}k$ is separable, and then the trace form $(x,y)\mapsto {\mathrm{Tr}}_{\kappa (p)\mathrm{/}k}(xy)$ is nondegenerate; the trace is $k$-linear (Every algebraic extension of a perfect field is separable, The trace form of a finite extension is nondegenerate exactly when the extension is separable, The norm $N_{K\mathrm{/}F}$ and trace ${\mathrm{Tr}}_{K\mathrm{/}F}$ of a finite field extension).

The Axiom of Choice is The Axiom of Choice.

The underlying space of $C$ is Noetherian: the finite-type affine-cover argument in Proper closed subsets of a curve are finite establishes this for the present integral finite-type scheme under Choice. For a Noetherian space, chain dimension is the supremum of the lengths of strict chains of nonempty irreducible closed subsets (Chain dimension and the empty-space convention). Since $C$ has chain dimension $1$, there is a strict chain $Z_0⊊Z_1$ of nonempty irreducible closed subsets of $C$.

Under the Axiom of Choice, every proper closed subset of an integral finite-type $k$-scheme of chain dimension $1$ is a finite set of closed points (Proper closed subsets of a curve are finite). The hypotheses hold for $C$.