Higher-dimensional duality is not imported into the curve theorem

This page records three concrete curve forms of Serre duality. The statements below retain their stated field hypotheses. Assume AC as in the cited curve theorems, and let $C$ be a smooth proper geometrically integral curve over a field $k$. All three forms use the same normalized trace $t_C:H^1(C,{\omega }_C)\to k$ (Dualizing line bundle and trace datum of a smooth projective variety).

1. For an invertible ${\mathcal{O}}_C$-module $\mathcal{L}$ over an arbitrary field $k$, there is the functorial perfect pairing $H^1(C,\mathcal{L})\times H^0(C,{\omega }_C\otimes {\mathcal{L}}^{-1})⟶k,(c,s)⟼t_C(c\cup s),$ where ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}^1$ (Serre duality for line bundles on a smooth proper curve, and the residue realization). The line-bundle theorem identifies this fixed trace with the negative of the positive residue-sum functional under its stated perfect-field hypothesis.
2. Over an arbitrary field, for a finite locally free ${\mathcal{O}}_C$-module $\mathcal{E}$ of rank $r$, there is the functorial perfect pairing $H^1(C,\mathcal{E})\times H^0(C,{\mathcal{E}}^{\vee }\otimes {\omega }_C)⟶k,(c,s)⟼t_C(c\cup s),$ using contraction and the same trace (Serre duality for finite locally free sheaves on a smooth proper curve). For rank one this is the line-bundle form in item 1.
3. For every coherent ${\mathcal{O}}_C$-module $\mathcal{F}$ over an arbitrary field, there are canonical, functorial isomorphisms ${\mathrm{Hom}}_{{\mathcal{O}}_C}(\mathcal{F},{\omega }_C)\cong H^1(C,\mathcal{F}{)}^{\ast },{\mathrm{Ext}}_{{\mathcal{O}}_C}^1(\mathcal{F},{\omega }_C)\cong H^0(C,\mathcal{F}{)}^{\ast }$ (Serre duality for coherent sheaves on a smooth proper curve, Ext form). Both are the pairings formed with the fixed trace. Explicitly, the first sends $\alpha :\mathcal{F}\to {\omega }_C$ to the functional $u\mapsto t_C(H^1(\alpha )(u))$. The second sends $\xi \in {\mathrm{Ext}}^1(\mathcal{F},{\omega }_C)$ to the functional $s\mapsto t_C({\chi }_{{\omega }_C}({\mathrm{Ext}}^1(s,{\omega }_C)(\xi )))$, where $s:{\mathcal{O}}_C\to \mathcal{F}$ is a global section and ${\chi }_{{\omega }_C}:{\mathrm{Ext}}^1({\mathcal{O}}_C,{\omega }_C)\cong H^1(C,{\omega }_C)$ is the canonical comparison used in that theorem.

The Ext groups in item 3 are global Ext as defined in Sheaf Ext of coherent modules: they are computed as the cohomology of ${\mathrm{Hom}}_{{\mathcal{O}}_C}(\mathcal{F},I^{\bullet })$ for an injective resolution of the second argument. The definition distinguishes these groups from the sheaves $\mathcal{E}x{t}^q$ and notes that, in positive degree, global Ext is not generally the global sections of sheaf Ext. The groups $H^q(C,-)$ are the right-derived functors of global sections (Sheaf cohomology as right derived global sections).

The coherent form is established on its own theorem page using an actual finite locally free resolution on the curve. For an ample invertible sheaf $L$, a sufficiently large twist $\mathcal{F}\otimes L^{\otimes m}$ is globally generated; its finite-dimensional space of global sections gives a surjection $\mathcal{E}\twoheadrightarrow \mathcal{F}$ with $\mathcal{E}$ finite locally free. Its kernel ${\mathcal{E}}^{\mathrm{\prime }}$ is coherent and torsion-free, hence finite locally free on the smooth curve. Thus the resolution used there is $0⟶{\mathcal{E}}^{\mathrm{\prime }}⟶\mathcal{E}⟶\mathcal{F}⟶0.$ This is the curve-specific argument in Serre duality for coherent sheaves on a smooth proper curve, Ext form; it does not invoke the projective-space finite-resolution lemma.

The line-bundle and finite locally free curve theorems specialize the published Serre duality theorem for finite locally free sheaves on a smooth projective variety over a field (Serre duality for locally free sheaves on a smooth projective variety). The coherent theorem builds on the finite locally free curve pairing and supplies the curve-specific resolution argument above. This summary adds no higher-dimensional duality claim. It does not import dualizing complexes, Grothendieck duality for non-proper or higher-dimensional morphisms, local cohomology, or relative duality over a base scheme more general than a field.