TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)audited 2026-10-02

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Functions on a proper curve

Statement

Assume the Axiom of Choice. Let $C$ be a proper curve over a field $k$ that is geometrically connected and geometrically reduced; for instance $C$ may be any smooth proper curve. Then the canonical map $k \to H^{0} (C, O_{C})$ , $c \mapsto c \cdot 1$ , is an isomorphism. More generally, for a proper integral curve over $k$ the $k$ -algebra $H^{0} (C, O_{C})$ is a finite field extension of $k$ ; it equals $k$ whenever $C$ is geometrically connected and geometrically reduced, and hence for every proper curve over $k$ , which is geometrically integral by definition.

Facts & Assumptions

Given: A field $k$ and a curve $C$ over $k$ whose structure morphism is proper; possibly also that $C$ is smooth over $k$ .

[F1]

A curve over $k$ is nonempty, geometrically integral, separated over $k$ , of finite type over $k$ , and of chain dimension one; a smooth curve is additionally smooth over $k$ . (Curves over a field)

[F2]

Geometric integrality of $X \to S$ means that the chosen algebraic-closure fibre is integral; an integral scheme is reduced and irreducible, and an irreducible space is connected. (Geometric properties of fibres)

[F3]

Under Choice, if $X$ is a nonempty scheme proper over $k$ whose algebraic-closure fibre is connected and reduced, then the unit map $k \to H^{0} (X, O_{X})$ is an isomorphism of $k$ -algebras. (Global functions on geometrically connected and geometrically reduced proper schemes)

[F4]

Under Choice, if $X$ is a nonempty proper integral finite-type $k$ -scheme with function field $K = k (X)$ , then $Γ (X, O_{X})$ is a finite field extension of $k$ contained in $K$ ; if $X$ is moreover geometrically integral over the chosen algebraic closure, then $Γ (X, O_{X}) = k$ . (Global functions on proper integral schemes form a finite extension of the base field)

[F5]

A morphism is proper if and only if it is separated, of finite type and universally closed; in particular a proper $k$ -scheme is of finite type over $k$ . (Proper morphisms)

[F6]

The degree-zero sheaf cohomology of $O_{X}$ is the group of global sections: $H^{0} (X, O_{X}) = Γ (X, O_{X})$ , with its $k$ -algebra structure. (Sheaf cohomology as right derived global sections)

[F7]

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

Proof

technique · direct; separate the geometric hypotheses from the bare integrality of the curve and invoke the two published global-functions results

1.1F1F5F6given

Let $C$ be a proper curve over $k$ . By [F1] the scheme $C$ is nonempty, reduced, irreducible, separated and of finite type over $k$ , and has chain dimension one; by [F5] it is proper and finite type over $k$ , hence a nonempty proper integral finite-type $k$ -scheme. Moreover $H^{0} (C, O_{C}) = Γ (C, O_{C})$ as $k$ -algebras [F6].

1.2F1F2given

If $C$ is a smooth curve, then it is in particular a curve, hence geometrically integral by [F1], so that its algebraic-closure fibre is integral by [F2], and an integral fibre is reduced and (being irreducible) connected. Thus the hypotheses of [F3] are satisfied for every smooth proper curve, and likewise for every proper curve, since curves are geometrically integral by definition.

1.3F4F5given

If $C$ is a proper integral curve over $k$ , then it is a nonempty proper integral finite-type $k$ -scheme by [F5], with function field $k (C)$ ; so [F4] applies to it.

2.1F3F7step 1.2

Under Choice [F7], [F3] together with the geometric hypotheses verified in step 1.2 shows that the unit map $k \to H^{0} (C, O_{C})$ is an isomorphism for every proper curve that is geometrically connected and geometrically reduced; this covers in particular every smooth proper curve and every proper curve.

2.2F1F4F6F7step 1.3

Under Choice [F7], [F4] with step 1.3 shows that for a proper integral curve $C$ the $k$ -algebra $Γ (C, O_{C}) = H^{0} (C, O_{C})$ is a finite field extension of $k$ contained in $k (C)$ , and that it equals $k$ as soon as $C$ is geometrically integral; since every curve is geometrically integral by [F1], this gives $H^{0} (C, O_{C}) ≅ k$ for every proper curve over $k$ .

3.1F1F3F4step 2.1step 2.2∎

The first sentence of the statement is step 2.1 with the smooth case supplied by step 1.2; the general assertion about proper integral curves is the first clause of step 2.2; the clause on geometrically connected and geometrically reduced curves is step 2.1, and the final clause on every proper curve is the geometric integrality of curves used in steps 1.2 and 2.2. The Axiom of Choice is used exactly through [F3] and [F4], both of which assume it.

Depends on

Used by

A degree-zero line bundle with a nonzero section is trivial Corollary
Finite morphisms from a curve to the projective line Corollary
Rational functions with poles bounded at one point Corollary
The canonical bundle has exactly g independent sections Corollary
The canonical divisor has degree 2g - 2 Corollary
The dimension of a complete linear system Corollary
Degree 2g does not force very ampleness Counterexample
The Riemann inequality is not an equality for special divisors Counterexample
Genus and arithmetic genus of a curve Definition
Genus via the Euler characteristic Definition
Hyperelliptic curves and hyperelliptic maps Definition
The Riemann-Roch dimension l(D) Definition
The empty divisor, its Euler characteristic and the genus boundary cases Example
A nonconstant rational function defines a finite map to the projective line Lemma
Arithmetic genus, geometric genus and delta invariants Lemma
Effective divisors linearly equivalent to D are sections modulo scalars Lemma
Normalization of the trace for Serre duality on a curve Remark
A base-point-free linear system defines a morphism to projective space Theorem
Serre duality for line bundles on a smooth proper curve, and the residue realization Theorem
The global residue theorem on a smooth proper curve over a perfect field Theorem

Dependency tree · two levels

102 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)
Ravi Vakil, The Rising Sea (version of October 21, 2025), Chs. 19 and 21 (standard reference, not scraped)