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Serre duality for finite locally free sheaves on a smooth proper curve

Statement

Assume the Axiom of Choice as inherited from the duality suppliers. Let $C$ be a smooth proper geometrically integral curve over a field $k$ and let $E$ be a finite locally free $O_{C}$ -module of rank $r$ . Then there is a functorial perfect $k$ -bilinear pairing $H^{1} (C, E) \times H^{0} (C, E^{\lor} \otimes ω_{C}) ⟶ k$ given by $c \cup s$ followed by the normalized trace $t_{C} ⁣ : H^{1} (C, ω_{C}) \to k$ , and consequently a canonical $k$ -linear isomorphism $H^{1} (C, E)^{*} ≅ H^{0} (C, E^{\lor} \otimes ω_{C})$ and the dimension identity $h^{1} (C, E) = h^{0} (C, E^{\lor} \otimes ω_{C})$ . For $E$ of rank one this recovers the line-bundle form of Serre duality for curves, and the result is stated over an arbitrary field with no perfectness hypothesis.

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; a smooth proper geometrically integral curve $C$ over $k$ ; and a finite locally free $O_{C}$ -module $E$ .

[F1]

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

[F2]

Every smooth proper geometrically integral curve $C$ over a field $k$ admits a closed immersion $i ⁣ : C \to P_{k}^{N}$ over $k$ , so that the structure morphism $C \to Spec k$ is projective in the H-projective convention (Every smooth proper curve admits a projective embedding).

[F3]

The canonical bundle of a smooth curve $C$ over $k$ is $ω_{C} = Ω_{C / k}^{1}$ ; since $C$ is smooth of relative dimension one over the field $k$ , this sheaf is locally free of rank one, so it is the sheaf $⋀^{1} Ω_{C / k}^{1}$ (Canonical bundle and canonical divisors).

[F4]

For a smooth projective $k$ -scheme $X$ of pure dimension $n$ one writes $ω_{X} : = \det Ω_{X / k}^{1} = ⋀^{n} Ω_{X / k}^{1}$ , and a normalized Serre trace for $X$ is a $k$ -linear map $t_{X} ⁣ : H^{n} (X, ω_{X}) \to k$ whose normalization is the Gysin compatibility $t_{X} = t_{P^{N}} \circ G_{j}$ along any closed immersion $j ⁣ : X ↪ P_{k}^{N}$ over $k$ ; the existence and embedding-independence of such a trace are claims of the duality theorem cited in [F6] (Dualizing line bundle and trace datum of a smooth projective variety).

[F5]

A finite locally free $O_{C}$ -module is a sheaf of $O_{C}$ -modules admitting a local isomorphism to $O_{C}^{r}$ , with locally constant rank function $r$ ; the dual sheaf is $E^{\lor} = H o m_{O_{C}} (E, O_{C})$ and $E^{\lor} \otimes_{O_{C}} ω_{C}$ is the sheaf tensor product constructed as the sheafification of the componentwise tensor presheaf (Locally free sheaves of finite rank, The internal Hom sheaf of two module sheaves, Tensor product of sheaves of modules).

[F6]

Let $X$ be a smooth projective $k$ -scheme of pure dimension $n$ and let $E$ be a finite locally free $O_{X}$ -module. With $ω_{X} = ⋀^{n} Ω_{X / k}^{1}$ there is a normalized trace $t_{X} ⁣ : H^{n} (X, ω_{X}) \to k$ , independent of a projective embedding, such that for every $0 \leq q \leq n$ the cup product, contraction and trace give a functorial perfect pairing of finite-dimensional $k$ -vector spaces $H^{q} (X, E) \times H^{n - q} (X, E^{\lor} \otimes ω_{X}) ⟶ H^{n} (X, ω_{X}) \overset{t_{X}}{\to} k;$ outside $0 \leq q \leq n$ the relevant cohomology groups vanish (Serre duality for locally free sheaves on a smooth projective variety).

Proof

Proof technique: direct; specialise the published smooth-projective duality theorem to a curve, where $n = 1$ and $q = 1$ .

1.1F2F3F4

By [F2] the curve $C$ is projective over $k$ ; it is smooth over $k$ and, being a curve, has underlying space of dimension one, so $X = C$ satisfies the hypotheses of the duality theorem [F6] with pure dimension $n = 1$ , and the canonical bundle $ω_{C} = Ω_{C / k}^{1}$ of [F3] is exactly the dualizing line bundle $⋀^{1} Ω_{C / k}^{1}$ of [F4].

1.2F5

The module $E$ is finite locally free of rank $r$ by hypothesis; its dual $E^{\lor}$ and the tensor product $E^{\lor} \otimes ω_{C}$ are the sheaves of [F5], and the latter is the module paired against $H^{1} (C, E)$ in the duality theorem.

2.1F6step 1.1step 1.2

Apply the duality theorem [F6] to $X = C$ , $n = 1$ , $E = E$ and $q = 1$ : the cup product, contraction and the normalized trace $t_{C} ⁣ : H^{1} (C, ω_{C}) \to k$ give a functorial perfect $k$ -bilinear pairing $H^{1} (C, E) \times H^{0} (C, E^{\lor} \otimes ω_{C}) \to k$ , and both $k$ -vector spaces are finite-dimensional.

3.1step 2.1

Perfectness of the pairing of step 2.1 says that the induced maps $H^{1} (C, E) \to H^{0} (C, E^{\lor} \otimes ω_{C})^{*}$ and $H^{0} (C, E^{\lor} \otimes ω_{C}) \to H^{1} (C, E)^{*}$ are bijective; dualising the first gives the canonical $k$ -linear isomorphism $H^{1} (C, E)^{*} ≅ H^{0} (C, E^{\lor} \otimes ω_{C})$ , and applying $\dim_{k}$ to either isomorphism gives $h^{1} (C, E) = h^{0} (C, E^{\lor} \otimes ω_{C})$ .

4.1F4F5step 3.1

Specialisation to rank one: if $r = 1$ then $E$ is finite locally free of rank one, so the displayed pairing is $H^{1} (C, E) \times H^{0} (C, E^{\lor} \otimes ω_{C}) \to k$ given by the cup product followed by the same normalized trace $t_{C}$ , which is the line-bundle form of Serre duality for curves; thus the present theorem generalises the invertible-coefficient case without changing the trace.

5.1F1F4F6step 4.1∎

No perfectness of $k$ was used: the duality theorem [F6] is stated over an arbitrary field, the projective embedding of step 1.1 and the canonical bundle identification exist over every field, and functoriality in $E$ together with embedding-independence of $t_{C}$ are parts of the cited theorem and definition; the only choice-theoretic input is the Axiom of Choice inherited through the duality suppliers [F1].

Depends on

Used by

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Sources

Ravi Vakil, The Rising Sea (version of October 21, 2025) (standard reference, not scraped)
Joseph Lipman, Residues, duality, and the fundamental class of a scheme-map (2011) (standard reference, not scraped)