Serre duality for line bundles on a smooth proper curve, and the residue realization

Given: the Axiom of Choice; a field $k$; a smooth proper geometrically integral curve $C$ over $k$; and an invertible ${\mathcal{O}}_C$-module $\mathcal{L}$.

The Axiom of Choice: every family of nonempty sets has a choice function (The Axiom of Choice).

Every smooth proper geometrically integral curve $C$ over a field $k$ admits a closed immersion over $k$ into ${\mathbf{P}}_k^N$, so $C\to \mathrm{Spec}k$ is projective in the H-projective convention (Every smooth proper curve admits a projective embedding).

The canonical bundle of a smooth curve is ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}^1={\bigwedge }^1{\mathrm{\Omega }}_{C\mathrm{/}k}^1$. The normalized trace in the published smooth-projective duality theorem is fixed by the Gysin map from an embedding and the Laurent-coefficient trace on projective space; embedding independence is part of that theorem. Thus the trace is not a freely chosen functional (Canonical bundle and canonical divisors, Dualizing line bundle and trace datum of a smooth projective variety, Serre duality for locally free sheaves on a smooth projective variety).

An invertible ${\mathcal{O}}_C$-module $\mathcal{L}$ is locally free of rank one; its dual ${\mathcal{L}}^{\vee }$ is again invertible, ${\mathcal{L}}^{\vee }\otimes \mathcal{L}\cong {\mathcal{O}}_C$ canonically, and ${\mathcal{L}}^{-1}$ denotes this dual. A rank-one locally free module is finite locally free of rank one (Invertible sheaves, Locally free sheaves of finite rank).

Tensor products of ${\mathcal{O}}_C$-modules are the sheafifications of the componentwise tensor presheaves, and the internal Hom is the sheaf of local Hom modules. For invertible sheaves these constructions are compatible with duals (Tensor product of sheaves of modules, The internal Hom sheaf of two module sheaves).

If $X$ is smooth projective of pure dimension $n$ over $k$ and $E$ is finite locally free, then ${\omega }_X={\bigwedge }^n{\mathrm{\Omega }}_{X\mathrm{/}k}^1$ has a normalized trace $t_X\text{\hspace{0.17em}⁣}:H^n(X,{\omega }_X)\to k$, independent of the chosen projective embedding, and cup product, contraction and trace give a functorial perfect pairing $H^q(X,E)\times H^{n-q}(X,E^{\vee }\otimes {\omega }_X)\to H^n(X,{\omega }_X)\stackrel{t_X}{\to }k$ for $0\le q\le n$ (Serre duality for locally free sheaves on a smooth projective variety).

For this geometrically integral proper curve, $H^0(C,{\mathcal{O}}_C)=k$. Applying [F6] to $E={\mathcal{O}}_C$ and $q=0$ gives ${\dim }_k{H}^1(C,{\omega }_C)=1$ and shows $t_C$ is nonzero (Functions on a proper curve).

Over a perfect field, the local coefficient-trace residue at each closed point is independent of the uniformizer and the residue pairing is well defined on cohomology and bilinear (Residue of a rational differential at a separable closed point, The residue pairing is well defined on cohomology).

Over a perfect field, the sum of local residues of a rational differential on $C$ is zero (The global residue theorem on a smooth proper curve over a perfect field). Together with the principal-parts presentation of $H^1$, this makes $t_C^{\mathrm{r}\mathrm{e}\mathrm{s}}(\xi )={\sum }_p{\mathrm{res}}_p({\xi }_p)$ a well-defined functional on $H^1(C,{\omega }_C)$ (H^1 of a line bundle on a curve as principal parts modulo meromorphic and regular sections).

In the local resolution $A{e}_t\stackrel{t}{\to }A$ of a rational point, the Cartier extension $0\to {\omega }_C\to {\omega }_C(p)\to \kappa (p)\to 0$ with quotient generator $t^{-1}dt$ has raw extension cocycle $e_t\mapsto +dt$. The normalized trace-one point class is $e_t\mapsto -dt$; the passage from the raw global extension class to that normalized class contributes the factor ${\sigma }_1=-1$ exactly once. Its point-to-curve-to-ambient Gysin trace is $+1$, so the original Cartier boundary has fixed trace $-1$. For a closed point with finite separable residue field, the twisting sequence and its connecting map commute with extension to an algebraic closure, and the fixed Gysin trace also commutes with that extension (Rational-point Koszul residue normalization for a smooth projective embedding, Local-to-global Ext collapse for a regular immersion, Embedding compatibility of smooth-projective Gysin traces, Flat field extension commutes with coherent cohomology).

The ordered Čech complex has differential $(\delta s{)}_{ij}=s_j{\mathrm{\mid }}_{U_i\cap U_j}-s_i{\mathrm{\mid }}_{U_i\cap U_j}$ for $i<j$, computes quasi-coherent cohomology on a finite affine cover of a separated scheme, and its comparison with sheaf cohomology is natural (Ordered Čech cochain complex of a cover, Cech cohomology computes quasi-coherent cohomology on a separated scheme, Canonical map from fixed-cover Čech to sheaf cohomology).

A nonempty finite-type curve has a closed point; its residue field is finite over $k$. If $k$ is perfect, that extension is separable. The cited curve-topology lemma gives that the underlying space of $C_K$ is Noetherian; by the definition of a scheme affine opens form a basis, and every open in a Noetherian space is quasi-compact (otherwise successive finite subunions of an open cover give a strictly increasing chain). Thus $C_K\setminus \{x\}$ has a finite affine cover. For a finite separable extension $L\mathrm{/}k$, the trace form is nondegenerate, so there is $a\in L$ with ${\mathrm{Tr}}_{L\mathrm{/}k}(a)\mathrm{\ne }0$ (Proper closed subsets of a curve are finite, Noetherian topological spaces via ACC on opens or DCC on closed subsets, Schemes, The trace form of a finite extension is nondegenerate exactly when the extension is separable).

If $L\mathrm{/}k$ is finite separable and $K=\bar{k}$, choose a primitive element $L=k[\alpha ]$. Its minimal polynomial factors over $K$ into distinct linear factors, and the Chinese remainder theorem gives $L{\otimes }_{k}K\cong {\prod }_{\sigma :L\hookrightarrow K}K$. The trace is the sum over these embeddings (A finite extension generated by elements all but possibly one of which are separable is simple, Norm and trace from embeddings, with the inseparable exponent in the norm formula).

Every class of $H^1(C,\mathcal{L})$ is represented by a finite-support family of local principal parts, equivalently by the connecting image of a section of $\mathcal{L}(D){\mathrm{\mid }}_D$ for some effective divisor $D$ containing those parts (H^1 of a line bundle on a curve as principal parts modulo meromorphic and regular sections).

For an effective Cartier point $x\hookrightarrow C$, the global Ext group ${\mathrm{Ext}}_{{\mathcal{O}}_C}^1(\kappa (x),{\omega }_C)$ is naturally identified with the stalk module Ext at $x$. Here is the needed support argument. Under AC, $\mathrm{Mod}({\mathcal{O}}_C)$ has enough injectives; restriction of injectives to opens is injective by exact extension by zero, and for an injective $I$ the sheaf $\mathcal{H}om(\kappa (x),I)$ is flasque: a morphism on an open $U$ transposes to $j_{!}(\kappa (x){\mathrm{\mid }}_U)\to I{\mathrm{\mid }}_V$ and extends across the monomorphism $j_{!}(\kappa (x){\mathrm{\mid }}_U)\hookrightarrow \kappa (x){\mathrm{\mid }}_V$ by injectivity. Thus it is $\mathrm{\Gamma }$-acyclic and the Grothendieck spectral sequence has $E_2^{p,q}=H^p(C,\mathcal{E}x{t}^q(\kappa (x),{\omega }_C))$ and abuts to global Ext. The locally free resolution $0\to {\mathcal{O}}_C(-x)\to {\mathcal{O}}_C\to \kappa (x)\to 0$ computes the sheaf Ext: its Hom complex into ${\omega }_C$ is ${\omega }_C\to {\omega }_C(x)$, so $\mathcal{E}x{t}^0=0$, $\mathcal{E}x{t}^1={\iota }_{\ast }({\omega }_C(x){\mathrm{\mid }}_x)$, and higher sheaf Ext vanishes. Its global sections are the local module Ext computed by $0\to {\mathcal{O}}_{C,x}\stackrel{\pi }{\to }{\mathcal{O}}_{C,x}\to \kappa (x)\to 0$, where $\pi$ is a local equation of the Cartier point. It is supported at the one affine point, so its positive cohomology vanishes. The spectral sequence therefore collapses in total degree one, and its edge is the natural isomorphism to the local module Ext. The sheaf-Ext computation can be checked directly with the same resolution: take an injective resolution ${\omega }_C\to I^{\bullet }$ and the double complex $\mathcal{H}om(P^{-r},I^s)$ for $P=[{\mathcal{O}}_C(-x)\to {\mathcal{O}}_C]$ in degrees $-1,0$. For each $s$, the augmented row $0\to \mathcal{H}om(\kappa (x),I^s)\to \mathcal{H}om({\mathcal{O}}_C,I^s)\to \mathcal{H}om({\mathcal{O}}_C(-x),I^s)\to 0$ is exact because $I^s$ is injective, also after restriction to opens. For each term $P^{-r}$, local freeness makes $0\to \mathcal{H}om(P^{-r},{\omega }_C)\to \mathcal{H}om(P^{-r},I^0)\to \mathcal{H}om(P^{-r},I^1)\to \cdots$ exact. The exact-row/column assembly lemmas identify the total cohomology with both that of $\mathcal{H}om(\kappa (x),I^{\bullet })$ and that of $\mathcal{H}om(P^{\bullet },{\omega }_C)$. The same extension-by-zero argument with source ${\mathcal{O}}_C$ makes $I\cong \mathcal{H}om({\mathcal{O}}_C,I)$ flasque, so the cited acyclic-resolution theorem computes sheaf cohomology by this injective resolution. No projective-space Ext-collapse statement is applied to $x\hookrightarrow C$. (Effective cartier divisor, Invertible sheaf of cartier divisor, Sheaf Ext of coherent modules, Sheaf cohomology as right derived global sections, Extension by zero for abelian sheaves on an open subspace, Flasque sheaf, Closed immersion preserves cohomology and coherent pushforward, Acyclic assembly by exact columns, Acyclic assembly by exact rows, Enough injective sheaves of modules, Effective Cartier divisors give a short exact sequence, The acyclic-resolution theorem for right derived functors, AC implies DC implies countable choice, Effective Cartier divisors are closed subschemes cut out by regular equations, Extension by zero is left adjoint to restriction and is exact on abelian sheaves, Flasque abelian sheaves are Γ-acyclic, Grothendieck spectral sequence, Local rings at closed points of smooth curves are discrete valuation rings, Affine acyclicity of quasi-coherent sheaves).

In the abelian category of ${\mathcal{O}}_C$-modules, with enough injectives as in [F15], the class of a short exact sequence is sent to its usual cone connecting morphism in the derived category; composing the extension $0\to {\omega }_C\to {\omega }_C(x)\to \kappa (x)\to 0$ with evaluation ${\mathcal{O}}_C\to \kappa (x)$ is exactly its long-exact-sequence boundary in ${\mathrm{Ext}}_{{\mathcal{O}}_C}^1({\mathcal{O}}_C,{\omega }_C)=H^1(C,{\omega }_C)$. The convention is the positive cone projection, with no additional shift scalar (Yoneda product is composition in the derived category).