LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02

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Euler characteristic changes by the residue degree

Statement

Assume the Axiom of Choice, inherited from the Euler-characteristic, finiteness and skyscraper suppliers below. Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field), let $D$ be a divisor on $C$ (Divisors on a smooth proper curve) and let $p \in C$ be a closed point with residue degree $d = [κ (p) : k]$ (Degree divisor proper curve). Then $χ (C, O_{C} (D + p)) = χ (C, O_{C} (D)) + d,$ and more generally $χ (C, O_{C} (D + E)) = χ (C, O_{C} (D)) + \deg_{k} (E)$ for every effective divisor $E \geq 0$ on $C$ , where $χ$ is the Euler characteristic of coherent sheaves on the proper $k$ -scheme $C$ (Euler characteristic of a coherent sheaf). Both identities hold in $Z$ .

The short exact sequence $0 \to O_{C} (D) \to O_{C} (D + p) \to i_{p, *} κ (p) \to 0$ , the cokernel $Q_{E}$ of $O_{C} (D) \to O_{C} (D + E)$ together with $\dim_{k} H^{0} (C, Q_{E}) = \deg_{k} (E)$ and the higher vanishing, and the coherence of $i_{p, *} κ (p)$ and of $Q_{E}$ are supplied by The exact sequence for adding one point to a divisor. That current supplier states and proves the Cartier-to-Weil, Cartier-sheaf and local-order interfaces used to construct the sequence; its Cartier-to-Weil route obtains Dependent Choice from the Axiom of Choice by AC implies DC implies countable choice. This corollary uses the stated sequence interface and does not certify its separate suppliers.

Facts & Assumptions

Given: the Axiom of Choice inherited from the Euler-characteristic, finiteness and skyscraper suppliers; a field $k$ , a smooth proper geometrically integral curve $C$ over $k$ , a divisor $D$ on $C$ , a closed point $p \in C$ and an effective divisor $E \geq 0$ on $C$ .

[F1]

The curve $C$ is proper and geometrically integral over the field $k$ , and $\deg_{k}$ is the group homomorphism on divisors with $\deg_{k} (D^{'}) = \sum_{x} n_{x} [κ (x) : k]$ (Curves over a field, Divisors on a smooth proper curve, Degree divisor proper curve, Effective divisors have nonnegative degree).

[F2]

For every divisor $D^{'}$ the sheaf $O_{C} (D^{'})$ is a coherent $O_{C}$ -module (Finite-dimensionality of the Riemann-Roch space, Coherent module sheaves).

[F3]

Adding one point: $0 \to O_{C} (D) \to O_{C} (D + p) \to i_{p, *} κ (p) \to 0$ is a short exact sequence of coherent $O_{C}$ -modules, $H^{0} (C, i_{p, *} κ (p)) ≅ κ (p)$ has $k$ -dimension $d = [κ (p) : k]$ while $H^{q} (C, i_{p, *} κ (p)) = 0$ for every $q \geq 1$ ; and for every effective divisor $E \geq 0$ the cokernel $Q_{E}$ of $O_{C} (D) \to O_{C} (D + E)$ is a coherent $O_{C}$ -module with $H^{q} (C, Q_{E}) = 0$ for every $q \geq 1$ and $\dim_{k} H^{0} (C, Q_{E}) = \deg_{k} (E)$ (The exact sequence for adding one point to a divisor).

[F4]

The Euler characteristic of a coherent module $F$ on a scheme $X$ proper over $k$ is $χ (X, F) = \sum_{q \geq 0} (- 1)^{q} \dim_{k} H^{q} (X, F)$ , a finite alternating sum of finite dimensions, and $χ (X, 0) = 0$ for the zero sheaf (Euler characteristic of a coherent sheaf, Sheaf cohomology as right derived global sections, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

[F5]

If $0 \to F^{'} \to F \to F^{''} \to 0$ is a short exact sequence of coherent $O_{X}$ -modules on a scheme $X$ proper over $k$ , then $χ (X, F) = χ (X, F^{'}) + χ (X, F^{''})$ (Euler characteristic is additive in short exact sequences).

[F6]

The Axiom of Choice is used exactly through the Euler-characteristic supplier [F4], the finiteness and coherence suppliers [F2] and the single-point sequence supplier [F3]; no further selection is made below (The Axiom of Choice).

Proof

technique · direct; apply additivity of the Euler characteristic to the single-point sequence, evaluate $\chi$ on the skyscraper through its cohomology, and repeat for the iterated cokernel

1.1F1F2F3

Set-up. By [F1] the curve $C$ is proper over $k$ , and by [F2] the sheaves $O_{C} (D)$ and $O_{C} (D + p)$ are coherent $O_{C}$ -modules; consequently all Euler characteristics below are those of [F4]. By [F3] the sequence $0 \to O_{C} (D) \to O_{C} (D + p) \to i_{p, *} κ (p) \to 0$ is a short exact sequence of coherent $O_{C}$ -modules with $H^{0} (C, i_{p, *} κ (p)) ≅ κ (p)$ of $k$ -dimension $d$ and $H^{q} (C, i_{p, *} κ (p)) = 0$ for every $q \geq 1$ .

1.2F2F3F4F5

The general effective shift. Let $E \geq 0$ be effective. By [F2] the sheaves $O_{C} (D)$ and $O_{C} (D + E)$ are coherent, and by [F3] the cokernel $Q_{E}$ of $O_{C} (D) \to O_{C} (D + E)$ is coherent with $H^{q} (C, Q_{E}) = 0$ for $q \geq 1$ and $\dim_{k} H^{0} (C, Q_{E}) = \deg_{k} (E)$ ; hence $χ (C, Q_{E}) = \dim_{k} H^{0} (C, Q_{E}) = \deg_{k} (E)$ by the definition of $χ$ in [F4]. Applying additivity [F5] to $0 \to O_{C} (D) \to O_{C} (D + E) \to Q_{E} \to 0$ gives $χ (C, O_{C} (D + E)) = χ (C, O_{C} (D)) + χ (C, Q_{E}) = χ (C, O_{C} (D)) + \deg_{k} (E)$ .

2.1F3F4step 1.1

The Euler characteristic of the skyscraper. By [F4] the Euler characteristic of the coherent module $i_{p, *} κ (p)$ on the proper $k$ -scheme $C$ is the alternating sum $\sum_{q \geq 0} (- 1)^{q} \dim_{k} H^{q} (C, i_{p, *} κ (p))$ ; by step 1.1 every term with $q \geq 1$ vanishes and the remaining term is $\dim_{k} H^{0} (C, i_{p, *} κ (p)) = \dim_{k} κ (p) = d$ . Hence $χ (C, i_{p, *} κ (p)) = d$ .

3.1F5step 1.1step 2.1

The one-point shift. Applying additivity [F5] to the short exact sequence of step 1.1 gives $χ (C, O_{C} (D + p)) = χ (C, O_{C} (D)) + χ (C, i_{p, *} κ (p)) = χ (C, O_{C} (D)) + d$ by step 2.1.

4.1F2F3F4F6step 3.1step 1.2∎

Conclusion and choice accounting. Step 3.1 gives the one-point identity and step 1.2 the identity for every effective divisor $E$ , both in $Z$ because they are alternating sums of finite dimensions and the degree $\deg_{k} (E)$ is an integer. The Axiom of Choice is used only through the suppliers recorded in [F6], namely the Euler-characteristic definition of [F4] and the finiteness, coherence and skyscraper suppliers of [F2] and [F3], which themselves inherit it; no further selection is made above.

Depends on

Used by

Every divisor is a finite signed sum of points Lemma

Dependency tree · two levels

101 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

William Fulton, Algebraic Curves (Internet Archive copy), Chs. 8 and 6 (standard reference, not scraped)
Michael Artin, MIT 18.721 Introduction to Algebraic Geometry (July 20, 2020 notes), Ch. 8 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (version of October 21, 2025), Chs. 18.5 and 21 (standard reference, not scraped)