Serre duality for coherent sheaves on a smooth proper curve, Ext form

By [F2], $C$ is a separated Noetherian scheme with Noetherian underlying space of dimension one. Hence [F8] gives $H^q(C,\mathcal{G})=0$ for every quasi-coherent $\mathcal{G}$ and every $q>1$; the sheaves used below are coherent or finite locally free, hence quasi-coherent. The sheaf ${\omega }_C$ is invertible by [F3].

The projective-embedding corollary gives a closed immersion $i:C\hookrightarrow {\mathbf{P}}_k^N$ (Every smooth proper curve admits a projective embedding). Put $L=i^{\ast }{\mathcal{O}}_{{\mathbf{P}}_k^N}(1)$. This is very ample by the relative definition and therefore ample by Relative very ampleness implies relative ampleness. The closed immersion makes $C$ projective over $k$ by Projective morphisms before Proj, in the convention of the global-generation lemma.

For any ${\mathcal{O}}_C$-module $\mathcal{G}$, fix an injective resolution $\mathcal{G}\to I^{\bullet }$. By [F11], ${\mathrm{Ext}}^q({\mathcal{O}}_C,\mathcal{G})\cong H^q(C,\mathcal{G})$ naturally. For finite locally free $\mathcal{E}$, ${\mathcal{E}}^{\vee }\otimes I^{\bullet }$ is an injective resolution of ${\mathcal{E}}^{\vee }\otimes \mathcal{G}$: exactness follows from [F12], and ${\mathcal{E}}^{\vee }\otimes -$ is right adjoint to the exact functor $\mathcal{E}\otimes -$, so it preserves injectives. Termwise $\mathrm{\Gamma }(C,{\mathcal{E}}^{\vee }\otimes I^{\bullet })=\mathrm{Hom}(\mathcal{E},I^{\bullet })$; its terms are flasque by [F11], so it computes cohomology. Thus ${\mathrm{Ext}}^q(\mathcal{E},\mathcal{G})\cong H^q(C,{\mathcal{E}}^{\vee }\otimes \mathcal{G})$ naturally, from global Ext itself and not global sections of sheaf Ext.

Write $t_C:H^1(C,{\omega }_C)\to k$ for the same fixed normalized trace as in [F13]. Define ${\mathrm{\Phi }}_F:\mathrm{Hom}(F,{\omega }_C)\to H^1(F{)}^{\ast }$ by ${\mathrm{\Phi }}_F(\alpha )(u)=t_C(H^1(\alpha )(u))$. By [F11], this is the Yoneda pairing of ${\mathrm{Ext}}^1({\mathcal{O}}_C,F)\cong H^1(F)$ with $\alpha$, followed by the fixed trace; it is canonical and natural contravariantly in $F$. For finite locally free $\mathcal{E}$, $\mathrm{Hom}(\mathcal{E},{\omega }_C)=H^0({\mathcal{E}}^{\vee }\otimes {\omega }_C)$ by [F12], so [F13] makes ${\mathrm{\Phi }}_E$ an isomorphism.

Define ${\mathrm{\Psi }}_F:{\mathrm{Ext}}^1(F,{\omega }_C)\to H^0(F{)}^{\ast }$ by ${\mathrm{\Psi }}_F(\xi )(s)=t_C({\chi }_{{\omega }_C}({\mathrm{Ext}}^1(s,{\omega }_C)(\xi )))$, where $s:{\mathcal{O}}_C\to F$ is the global section and ${\chi }_{{\omega }_C}:{\mathrm{Ext}}^1({\mathcal{O}}_C,{\omega }_C)\to H^1(C,{\omega }_C)$ is [F11]. This is the Yoneda pairing followed by the fixed trace, and is canonical and natural contravariantly in $F$.

Choose $n\ge 0$ so that $\mathcal{F}(n):=\mathcal{F}\otimes L^{\otimes n}$ is globally generated, using [F6]. This twist is coherent. By [F7], $H^0(C,\mathcal{F}(n))$ is finite-dimensional over $k$; choose a finite $k$-basis $s_1,\dots ,s_N$.

For finite locally free $\mathcal{E}$, step 1.3 identifies ${\mathrm{Ext}}^1(E,{\omega }_C)$ with $H^1(E^{\vee }\otimes {\omega }_C)$; pullback along $s:{\mathcal{O}}_C\to E$ is induced by contraction $E^{\vee }\otimes {\omega }_C\to {\omega }_C$. The vector-bundle pairing [F13] for $E^{\vee }\otimes {\omega }_C$, whose dual bundle tensored with ${\omega }_C$ is canonically $E$, is exactly ${\mathrm{\Psi }}_E$. Hence ${\mathrm{\Psi }}_E$ and ${\mathrm{\Psi }}_{E^{\mathrm{\prime }}}$ are isomorphisms.

The evaluation map from all global sections is surjective by [F6]. Every global section is a $k$-linear combination of the $s_i$, with $k$ acting through ${\mathcal{O}}_C$; hence at each stalk the values of the $s_i$ generate $\mathcal{F}(n)$ over ${\mathcal{O}}_C$. Thus ${\mathcal{O}}_C^{\oplus N}\twoheadrightarrow \mathcal{F}(n)$, and after twisting by $L^{-n}$ there is a surjection $p:\mathcal{E}:=(L^{-n}{)}^{\oplus N}\twoheadrightarrow \mathcal{F}$ with $\mathcal{E}$ finite locally free.

Let ${\mathcal{E}}^{\mathrm{\prime }}:=\ker p$. It is coherent by [F5], and as a subsheaf of $\mathcal{E}$ it is torsion-free; [F4] makes it finite locally free. We obtain a two-term finite locally free resolution for every coherent $\mathcal{F}$, including torsion sheaves: $0⟶{\mathcal{E}}^{\mathrm{\prime }}\stackrel{\mathrm{ȷ}}{\to }\mathcal{E}\stackrel{p}{\to }\mathcal{F}⟶0.$

By [F9] and [F8], this short exact sequence gives the cohomology sequence below; it is exact at the final term because $H^2(C,{\mathcal{E}}^{\mathrm{\prime }})=0$. Every term is finite-dimensional by [F7]: $0\to H^0(E^{\mathrm{\prime }})\to H^0(E)\to H^0(F)\stackrel{{\delta }_H}{\to }{H}^1(E^{\mathrm{\prime }})\to H^1(E)\to H^1(F)\to 0,$ where $H^q(G)$ abbreviates $H^q(C,\mathcal{G})$.

With $\mathcal{G}={\omega }_C$, the exact first-variable Ext sequence supplied by [F10] is $0\to \mathrm{Hom}(F,{\omega }_C)\to \mathrm{Hom}(E,{\omega }_C)\to \mathrm{Hom}(E^{\mathrm{\prime }},{\omega }_C)\stackrel{{\delta }_{\mathrm{E}\mathrm{x}\mathrm{t}}}{\to }{\mathrm{Ext}}^1(F,{\omega }_C)\to {\mathrm{Ext}}^1(E,{\omega }_C)\to {\mathrm{Ext}}^1(E^{\mathrm{\prime }},{\omega }_C)$.

Dualizing the finite-dimensional cohomology tail $H^1(E^{\mathrm{\prime }})\to H^1(E)\to H^1(F)\to 0$ gives $0\to H^1(F{)}^{\ast }\to H^1(E{)}^{\ast }\to H^1(E^{\mathrm{\prime }}{)}^{\ast }$. The Ext sequence in 5.2 identifies $\mathrm{Hom}(F,{\omega }_C)$ with the kernel of $\mathrm{Hom}(E,{\omega }_C)\to \mathrm{Hom}(E^{\mathrm{\prime }},{\omega }_C)$. Naturality of $\mathrm{\Phi }$ for both $p:E\to F$ and $\mathrm{ȷ}:E^{\mathrm{\prime }}\to E$ identifies this kernel map with the dual cohomology kernel map; since ${\mathrm{\Phi }}_E$ and ${\mathrm{\Phi }}_{E^{\mathrm{\prime }}}$ are isomorphisms by step 1.4, the induced map on kernels, precisely ${\mathrm{\Phi }}_F$, is an isomorphism.

Dualizing the cohomology sequence of 5.1 gives the exact row $H^1(E{)}^{\ast }\to H^1(E^{\mathrm{\prime }}{)}^{\ast }\to H^0(F{)}^{\ast }\to H^0(E{)}^{\ast }\to H^0(E^{\mathrm{\prime }}{)}^{\ast }$. Together with 5.2 the five-lemma diagram is the following; its vertical maps, in order, are ${\mathrm{\Phi }}_E,{\mathrm{\Phi }}_{E^{\mathrm{\prime }}},{\mathrm{\Psi }}_F,{\mathrm{\Psi }}_E,{\mathrm{\Psi }}_{E^{\mathrm{\prime }}}$: [F7, F10, step 1.4, step 1.5, step 2.2, step 5.1, step 5.2] $\begin{array}{ccccccccc}\mathrm{Hom}(E,{\omega }_C)& \to & \mathrm{Hom}(E^{\mathrm{\prime }},{\omega }_C)& \to & {\mathrm{Ext}}^1(F,{\omega }_C)& \to & {\mathrm{Ext}}^1(E,{\omega }_C)& \to & {\mathrm{Ext}}^1(E^{\mathrm{\prime }},{\omega }_C)\\ \downarrow {\mathrm{\Phi }}_E& & \downarrow {\mathrm{\Phi }}_{E^{\mathrm{\prime }}}& & \downarrow {\mathrm{\Psi }}_F& & \downarrow {\mathrm{\Psi }}_E& & \downarrow {\mathrm{\Psi }}_{E^{\mathrm{\prime }}}\\ H^1(E{)}^{\ast }& \to & H^1(E^{\mathrm{\prime }}{)}^{\ast }& \to & H^0(F{)}^{\ast }& \to & H^0(E{)}^{\ast }& \to & H^0(E^{\mathrm{\prime }}{)}^{\ast }\mathrm{.}\end{array}$ The square over $\mathrm{Hom}(E)\to \mathrm{Hom}(E^{\mathrm{\prime }})$ commutes by naturality of $\mathrm{\Phi }$, and the squares over ${\mathrm{Ext}}^1(F)\to {\mathrm{Ext}}^1(E)\to {\mathrm{Ext}}^1(E^{\mathrm{\prime }})$ commute by naturality of $\mathrm{\Psi }$.

Let $\alpha :E^{\mathrm{\prime }}\to {\omega }_C$ and $s:{\mathcal{O}}_C\to F$. In the injective resolution ${\omega }_C\to I^{\bullet }$, extend $j\alpha :E^{\mathrm{\prime }}\to I^0$ to $\tilde{\alpha }:E\to I^0$, with $j:{\omega }_C\to I^0$ the coaugmentation. The connecting class ${\delta }_{\mathrm{E}\mathrm{x}\mathrm{t}}(\alpha )$ is represented by $d^0\tilde{\alpha }:E\to I^1$, which vanishes on $E^{\mathrm{\prime }}$ and factors as $c\circ p$ for a cocycle $c:F\to I^1$. The pushout $P=({\omega }_C\oplus E)\mathrm{/}\{(\alpha (e^{\mathrm{\prime }}),-\mathrm{ȷ}(e^{\mathrm{\prime }})):e^{\mathrm{\prime }}\in E^{\mathrm{\prime }}\}$ maps to $Q_c:=\{(x,z)\in F\oplus I^0:c(x)=d^{0}z\}$ by $[w,e]\mapsto (p(e),j(w)+\tilde{\alpha }(e))$; this is well-defined and induces the identity on kernel ${\omega }_C$ and quotient $F$, so its Yoneda class is the cocycle class ${\delta }_{\mathrm{E}\mathrm{x}\mathrm{t}}(\alpha )$ with positive sign. Precomposition by $s$ sends $c$ to $c\circ s$, the cocycle of the pullback extension. Under [F11], this Yoneda class maps to the cohomology boundary of $1$: for an extension $0\to G\to P_s\to {\mathcal{O}}_C\to 0$, extend $G\to I^0$ to $b:P_s\to I^0$; $d^{0}b$ factors through ${\mathcal{O}}_C$ and represents both the injective-resolution Ext class and, since $I^{\bullet }$ is flasque, the cohomology boundary. Thus the comparison introduces no sign. Naturality of the cohomology long exact sequence for the pushout diagram gives ${\chi }_{{\omega }_C}({\mathrm{Ext}}^1(s,{\omega }_C)({\delta }_{\mathrm{E}\mathrm{x}\mathrm{t}}(\alpha )))=H^1(\alpha )({\delta }_H(s))$. Equivalently, local lifts $e_i$ of $s(1)$ give differences $e_j-e_i$; in $P$ these become $[0,e_j-e_i]=[\alpha (e_j-e_i),0]$, again with positive sign. Applying $t_C$ verifies the middle square.

By [F14], the five lemma applies to the exact rows in 6.2: the outer vertical maps are isomorphisms by steps 1.4 and 2.2, while the explicitly defined middle map is ${\mathrm{\Psi }}_F$. Thus ${\mathrm{\Psi }}_F$ is an isomorphism. No vanishing of ${\mathrm{Ext}}^2(F,{\omega }_C)$ is needed.

Both maps are composition with the fixed normalized trace and are independent of the chosen resolution, which is used only to establish bijectivity. The first isomorphism gives the stated dimension identity, and for finite locally free $F$ it is vector-bundle duality by its defining pairing. The field is arbitrary; no perfectness or extra qualifier is introduced.