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Every divisor is a finite signed sum of points

Statement

Assume the Axiom of Choice, inherited from the Euler-characteristic suppliers below. Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) and let $D = \sum_{x} n_{x} [x]$ be a divisor on $C$ (Divisors on a smooth proper curve). Write $D^{+}$ and $D^{-}$ for the positive and negative parts of $D$ (Divisor support positive negative parts).

(Decomposition.) $D$ is a finite $Z$ -linear combination of closed points, and $D = D^{+} - D^{-}$ with $D^{+}$ and $D^{-}$ effective divisors of disjoint support; consequently $\deg_{k} (D) = \deg_{k} (D^{+}) - \deg_{k} (D^{-})$ (Degree divisor proper curve, Effective divisors have nonnegative degree).
(Chain.) Let $p_{1}, \dots, p_{m}$ be a listing of the points of $Supp (D^{+})$ in which each point $x$ occurs exactly $n_{x}$ times, and let $q_{1}, \dots, q_{n}$ be a listing of the points of $Supp (D^{-})$ in which each point $y$ occurs exactly $- n_{y}$ times. For every ordering of the $m + n$ signed symbols $+ [p_{1}], \dots, + [p_{m}], - [q_{1}], \dots, - [q_{n}]$ , the chain of divisors $M_{0} = 0$ , $M_{k} = M_{k - 1} \pm [r_{k}]$ , where the $k$ -th symbol is $\pm [r_{k}]$ , has successive differences $\pm [r_{k}]$ at closed points and ends at $M_{m + n} = \sum_{i} [p_{i}] - \sum_{j} [q_{j}] = D$ .
(Order-independent iterated computation.) Computing $χ$ along this chain by the one-point shift of Euler characteristic changes by the residue degree, every step changes the value by $+ [κ (r_{k}) : k]$ for an addition and by $- [κ (r_{k}) : k]$ for a removal, so the telescoping total is $χ (C, O_{C} (D)) = χ (C, O_{C} (0)) + \sum_{i} [κ (p_{i}) : k] - \sum_{j} [κ (q_{j}) : k] = χ (C, O_{C} (0)) + \deg_{k} (D^{+}) - \deg_{k} (D^{-}) = χ (C, O_{C} (0)) + \deg_{k} (D),$ a value that depends only on the multiset of listed points and not on the chosen ordering; identifying $O_{C} (0) ≅ O_{C}$ with the structure sheaf, this reads $χ (C, O_{C} (D)) = χ (C, O_{C}) + \deg_{k} (D)$ . Here $χ$ is the Euler characteristic of coherent sheaves on the proper $k$ -scheme $C$ (Euler characteristic of a coherent sheaf), and every sheaf appearing is coherent (Finite-dimensionality of the Riemann-Roch space).

The attachment of the invertible sheaf $O_{C} (D)$ to the divisor $D$ and the identification $O_{C} (0) ≅ O_{C}$ use the current interfaces of Invertible sheaf of cartier divisor and Cartier and Weil divisors agree on a smooth curve. The latter has an explicit Dependent Choice premise, supplied by the stated Axiom of Choice through AC implies DC implies countable choice; these premises are recorded in [F6] and [F7]. (Scaffold repair: the scaffold's phrase "enumeration of the support" is read as a listing with repetitions, each point occurring exactly as often as its coefficient, which is what makes the chain end at $D$ ; the statement above says this explicitly.)

Facts & Assumptions

Given: the Axiom of Choice inherited from the Euler-characteristic and Cartier-divisor suppliers; a field $k$ , a smooth proper geometrically integral curve $C$ over $k$ , a divisor $D = \sum_{x} n_{x} [x]$ on $C$ , and listings $p_{1}, \dots, p_{m}$ , $q_{1}, \dots, q_{n}$ as in part 2.

[F1]

Divisors, parts and degree. The divisors on $C$ are the finite formal integral combinations of closed points and form the free abelian group $Div (C)$ ; the support $Supp (D) = {x : n_{x} \neq 0}$ is finite, $D^{+} = \sum_{x} \max (n_{x}, 0) [x]$ , $D^{-} = \sum_{x} \max (- n_{x}, 0) [x]$ are effective with disjoint supports, and $D = D^{+} - D^{-}$ ; the $k$ -degree is $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ and $\deg_{k} : Div (C) \to Z$ is a group homomorphism, while $\deg_{k} (E) \geq 0$ for every effective $E \geq 0$ (Divisors on a smooth proper curve, Divisor support positive negative parts, Degree divisor proper curve, Effective divisors have nonnegative degree).

[F2]

The one-point shift. For every divisor $M$ on $C$ and every closed point $r \in C$ , $χ (C, O_{C} (M + [r])) = χ (C, O_{C} (M)) + [κ (r) : k]$ , and more generally $χ (C, O_{C} (M + E)) = χ (C, O_{C} (M)) + \deg_{k} (E)$ for every effective divisor $E \geq 0$ ; both identities hold in $Z$ (Euler characteristic changes by the residue degree).

[F3]

Coherence and finiteness. For every divisor $M$ on $C$ the invertible sheaf $O_{C} (M)$ is a coherent $O_{C}$ -module, and $H^{q} (C, O_{C} (M))$ is a finite-dimensional $k$ -vector space for every $q \geq 0$ that vanishes for $q \geq 2$ (Finite-dimensionality of the Riemann-Roch space).

[F4]

The Euler characteristic of a coherent module $F$ on the proper $k$ -scheme $C$ is $χ (C, F) = \sum_{q \geq 0} (- 1)^{q} \dim_{k} H^{q} (C, F)$ , a finite alternating sum of finite dimensions and hence an element of $Z$ (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]

Functoriality in the sheaf. For each $q \geq 0$ the assignment $F \mapsto H^{q} (C, F)$ is a covariant additive functor on abelian sheaves on $C$ , so a morphism $φ : F \to G$ induces $H^{q} (C, φ) : H^{q} (C, F) \to H^{q} (C, G)$ compatibly with identities and composites (Variance of sheaf cohomology); a functor sends isomorphisms to isomorphisms (Every functor preserves isomorphisms), so isomorphic sheaves have isomorphic cohomology groups in every degree and equal Euler characteristics.

[F6]

The current Cartier-to-Weil interface identifies the Weil divisors of this smooth proper curve with Cartier divisors and preserves their principal divisors (Cartier and Weil divisors agree on a smooth curve). The associated sheaf is constructed with $O_{C} (D) \subseteq K_{C}$ and $O_{C} (0) ≅ O_{C}$ by Invertible sheaf of cartier divisor. These are the interfaces for every sheaf $O_{C} (M)$ in the chain.

[F7]

The Axiom of Choice is used through the Euler-characteristic supplier [F2], the finiteness and coherence supplier [F3], the Euler-characteristic definition [F4] and the Cartier-to-Weil interface [F6]. In ZF, AC implies DC by AC implies DC implies countable choice, so the DC premise of Cartier and Weil divisors agree on a smooth curve is available from the stated assumption; no further selection is made below (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

technique · direct; split the divisor into positive and negative parts, expand each coefficient into that many single-point operations, walk the resulting finite chain through the divisor group while telescoping the one-point shifts of $\chi$, and observe that the total is a signed sum of residue degrees over a multiset that does not depend on the ordering

1.1F1

Decomposition. By [F1] the support of $D = \sum_{x} n_{x} [x]$ is a finite set of closed points, so $D$ is a finite $Z$ -linear combination of closed points. By [F1] the parts $D^{+} = \sum_{x} \max (n_{x}, 0) [x]$ and $D^{-} = \sum_{x} \max (- n_{x}, 0) [x]$ are effective divisors with disjoint supports and $D = D^{+} - D^{-}$ . Since $\deg_{k}$ is a group homomorphism [F1], $\deg_{k} (D) = \deg_{k} (D^{+}) - \deg_{k} (D^{-})$ , and both terms are nonnegative integers because $D^{+}$ and $D^{-}$ are effective [F1].

2.1F1step 1.1

The chain. In the free abelian group $Div (C)$ of [F1] one has $\sum_{i} [p_{i}] = \sum_{x \in Supp (D^{+})} n_{x} [x] = D^{+}$ and $\sum_{j} [q_{j}] = \sum_{y \in Supp (D^{-})} (- n_{y}) [y] = D^{-}$ , because the listings repeat each point with its coefficient; hence $\sum_{i} [p_{i}] - \sum_{j} [q_{j}] = D^{+} - D^{-} = D$ by step 1.1. Define $M_{0} = 0$ and, for $k = 1, \dots, m + n$ , $M_{k} = M_{k - 1} + [r_{k}]$ if the $k$ -th symbol is $+ [r_{k}]$ and $M_{k} = M_{k - 1} - [r_{k}]$ if it is $- [r_{k}]$ ; each $M_{k}$ is an element of $Div (C)$ , each successive difference $M_{k} - M_{k - 1}$ is $\pm [r_{k}]$ at the closed point $r_{k}$ , and the final term is $M_{m + n} = \sum_{i} [p_{i}] - \sum_{j} [q_{j}] = D$ for every ordering, because addition in the abelian group $Div (C)$ is commutative and associative.

3.1F2F3F4step 2.1

The shift at each step. Let $M \in Div (C)$ and let $r \in C$ be a closed point. Applying the one-point identity of [F2] to $M$ gives $χ (C, O_{C} (M + [r])) = χ (C, O_{C} (M)) + [κ (r) : k]$ , and applying it to $M - [r]$ in place of $M$ gives $χ (C, O_{C} (M)) = χ (C, O_{C} (M - [r])) + [κ (r) : k]$ , that is, $χ (C, O_{C} (M - [r])) = χ (C, O_{C} (M)) - [κ (r) : k]$ . All these values are defined: for every divisor $M^{'}$ the sheaf $O_{C} (M^{'})$ is coherent by [F3], so [F4] applies to the proper $k$ -scheme $C$ . Consequently each step of the chain of step 2.1 changes the Euler characteristic by $+ [κ (r_{k}) : k]$ for a symbol $+ [r_{k}]$ and by $- [κ (r_{k}) : k]$ for a symbol $- [r_{k}]$ .

4.1F1step 1.1step 2.1step 3.1

Telescoping. Induction on $k$ using step 3.1 gives $χ (C, O_{C} (M_{k})) = χ (C, O_{C} (0)) + \sum_{j = 1}^{k} ε_{j} [κ (r_{j}) : k]$ , where $ε_{j} = 1$ if the $j$ -th symbol is an addition and $ε_{j} = - 1$ if it is a removal. At $k = m + n$ this is $χ (C, O_{C} (D)) = χ (C, O_{C} (0)) + \sum_{i} [κ (p_{i}) : k] - \sum_{j} [κ (q_{j}) : k]$ by step 2.1, and the two sums are $\sum_{x \in Supp (D^{+})} n_{x} [κ (x) : k] = \deg_{k} (D^{+})$ and $\sum_{y \in Supp (D^{-})} (- n_{y}) [κ (y) : k] = \deg_{k} (D^{-})$ by the definition of the listings, so $χ (C, O_{C} (D)) = χ (C, O_{C} (0)) + \deg_{k} (D^{+}) - \deg_{k} (D^{-}) = χ (C, O_{C} (0)) + \deg_{k} (D)$ by step 1.1. The multiset of signed residue degrees ${ε_{j} [κ (r_{j}) : k]}$ is determined by the listings alone, so this total, and hence the iterated value, is independent of the chosen ordering.

5.1F2F3F4F5F6F7step 1.1step 2.1step 4.1∎

The base term and conclusion. The chain of step 2.1 starts at the zero divisor $0$ , whose attached sheaf is $O_{C} (0)$ ; the current dictionary [F6] identifies $O_{C} (0) ≅ O_{C}$ with the structure sheaf, and by [F5] isomorphic sheaves have isomorphic cohomology in every degree, hence equal Euler characteristics, so $χ (C, O_{C} (0)) = χ (C, O_{C})$ . With step 4.1 this gives the asserted identity $χ (C, O_{C} (D)) = χ (C, O_{C}) + \deg_{k} (D)$ . Steps 1.1, 2.1 and 4.1 prove the decomposition, chain and order-independence clauses, so all three parts of the Statement hold. The Axiom of Choice is used only through the suppliers recorded in [F7], namely the one-point shift [F2], the coherence and finiteness of [F3], the Euler-characteristic definition [F4] and the current dictionary [F6], with DC supplied by AC as recorded there; no further selection is made, the listings of part 2 being finite and fixed.

Depends on

Used by

Riemann-Roch in Euler-characteristic form: the degree shift Theorem

Dependency tree · two levels

100 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)
The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)