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

Statement

Assume the Axiom of Choice, inherited from the Euler-characteristic and finiteness suppliers below. Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) and let $D$ be a divisor on $C$ (Divisors on a smooth proper curve). Then $χ (C, O_{C} (D)) - χ (C, O_{C}) = \deg_{k} (D),$ an identity in $Z$ , and equivalently $h^{0} (D) - h^{1} (D) = \deg_{k} (D) + h^{0} (0) - h^{1} (0),$ where $h^{i} (D) = \dim_{k} H^{i} (C, O_{C} (D)) \geq 0$ is the notation of The Riemann-Roch dimension l(D) and $χ$ is the Euler characteristic of coherent sheaves on the proper $k$ -scheme $C$ (Euler characteristic of a coherent sheaf), a finite alternating sum of finite dimensions. No Serre duality is used: both forms leave the index of speciality $h^{1} (D)$ as an unknown nonnegative integer.

The attachment of $O_{C} (D)$ , the identity $O_{C} (0) ≅ O_{C}$ , and the global-section identification use the current Cartier-divisor interfaces Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors and Cartier and Weil divisors agree on a smooth curve, inherited through Every divisor is a finite signed sum of points.

Facts & Assumptions

Given: a field $k$ , a smooth proper geometrically integral curve $C$ over $k$ , and a divisor $D$ on $C$ .

[F1]

The curve $C$ is proper, separated and of finite type over the field $k$ , geometrically integral and of chain dimension one; a divisor on $C$ is a finite formal integral combination of closed points, $D = \sum_{x} n_{x} [x]$ , and the $k$ -degree is $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ , a group homomorphism $Div (C) \to Z$ (Curves over a field, Divisors on a smooth proper curve, Degree divisor proper curve).

[F2]

The decomposition lemma: $D = D^{+} - D^{-}$ with $D^{+}, D^{-}$ effective of disjoint support and $\deg_{k} (D) = \deg_{k} (D^{+}) - \deg_{k} (D^{-})$ ; for every listing of the points of $Supp (D^{+})$ repeated with their coefficients, followed by the points of $Supp (D^{-})$ repeated with the coefficients of $D^{-}$ , and for every ordering of the resulting signed symbols, the chain $M_{0} = 0$ , $M_{k} = M_{k - 1} \pm [r_{k}]$ ends at $D$ , and telescoping the one-point shift gives $χ (C, O_{C} (D)) = χ (C, O_{C} (0)) + \deg_{k} (D)$ , a value independent of the ordering; identifying $O_{C} (0) ≅ O_{C}$ with the structure sheaf this reads $χ (C, O_{C} (D)) = χ (C, O_{C}) + \deg_{k} (D)$ (Every divisor is a finite signed sum of points).

[F3]

Coherence, finiteness and the dimension form of $χ$ : for every divisor $D^{'}$ on $C$ the invertible sheaf $O_{C} (D^{'})$ is a coherent $O_{C}$ -module, $H^{q} (C, O_{C} (D^{'}))$ is a finite-dimensional $k$ -vector space for every $q \geq 0$ and vanishes for every $q \geq 2$ , and the Euler characteristic satisfies $χ (C, O_{C} (D^{'})) = h^{0} (D^{'}) - h^{1} (D^{'})$ (Finite-dimensionality of the Riemann-Roch space, 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).

[F4]

Notation: $h^{i} (D^{'}) = \dim_{k} H^{i} (C, O_{C} (D^{'}))$ for every $i \geq 0$ and divisor $D^{'}$ , so in particular $h^{0} (D^{'})$ and $h^{1} (D^{'})$ are the dimensions appearing in [F3], and $l (D^{'}) = h^{0} (D^{'})$ (The Riemann-Roch dimension l(D)).

[F5]

The current Cartier-divisor interfaces Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors and Cartier and Weil divisors agree on a smooth curve attach $O_{C} (D)$ to $D$ , identify $L (D)$ with $H^{0} (C, O_{C} (D))$ , and give $O_{C} (0) ≅ O_{C}$ . The proof uses the last identification for its base term.

[F6]

The Axiom of Choice is used exactly through the decomposition lemma [F2], the finiteness and coherence supplier [F3] and the flagged dictionary [F5]; no further selection is made below (The Axiom of Choice).

Proof

technique · direct; read the degree shift off the point-by-point decomposition of the divisor, convert both Euler characteristics into their $h^0-h^1$ forms, and compare the resulting identities to obtain the two displayed forms and their equivalence

1.1F1F3F4F5

Set-up. By [F1] the curve $C$ is proper over $k$ , so the Euler characteristic of [F3] is defined for coherent sheaves on $C$ . By [F3] the sheaves $O_{C} (D)$ and $O_{C} (0)$ are coherent $O_{C}$ -modules, so $χ (C, O_{C} (D))$ , $χ (C, O_{C} (0))$ and, through the flagged dictionary [F5], $χ (C, O_{C}) = χ (C, O_{C} (0))$ are all integers; by [F4] the symbols $h^{i} (D)$ are the dimensions of [F3].

1.2F2F5

The degree shift. By part 3 of the decomposition lemma [F2], applied to the divisor $D$ , the telescoping of the one-point shifts along any ordering of the listed signed points gives $χ (C, O_{C} (D)) = χ (C, O_{C} (0)) + \deg_{k} (D)$ ; by the flagged identification $O_{C} (0) ≅ O_{C}$ of [F5] this reads $χ (C, O_{C} (D)) = χ (C, O_{C}) + \deg_{k} (D)$ . Rearranging in $Z$ gives the first displayed identity $χ (C, O_{C} (D)) - χ (C, O_{C}) = \deg_{k} (D)$ ; the ordering-independence asserted in [F2] shows that the value does not depend on how the listing is traversed.

2.1F3F4step 1.2

The dimension forms of the two Euler characteristics. By [F3] applied to the divisor $D$ , $χ (C, O_{C} (D)) = h^{0} (D) - h^{1} (D)$ ; by [F3] applied to the zero divisor, $χ (C, O_{C} (0)) = h^{0} (0) - h^{1} (0)$ . Substituting the second identity into the un-flagged form of step 1.2 gives $h^{0} (D) - h^{1} (D) = \deg_{k} (D) + h^{0} (0) - h^{1} (0)$ , the second displayed identity; no Serre duality is involved, the terms $h^{1} (D)$ and $h^{1} (0)$ being the nonnegative dimensions of [F3] and [F4].

3.1F3F5step 1.2step 2.1

Equivalence of the two forms. Assume first the first displayed identity. By step 2.1, $h^{0} (D) - h^{1} (D) = χ (C, O_{C} (D))$ and $h^{0} (0) - h^{1} (0) = χ (C, O_{C} (0))$ ; by the flagged dictionary [F5], $χ (C, O_{C} (0)) = χ (C, O_{C})$ , so the first identity rearranges to the second. Conversely, assume the second displayed identity; then by step 2.1, $χ (C, O_{C} (D)) - χ (C, O_{C} (0)) = \deg_{k} (D)$ , and by the flagged dictionary [F5] the second term is $χ (C, O_{C})$ , giving the first identity. Thus the two displayed forms are equivalent under the flagged identification, and each of them is proved: the first in step 1.2 with the flag, the second in step 2.1 without it.

4.1F2F3F5F6step 1.2step 2.1step 3.1∎

Conclusion and choice accounting. Step 1.2 gives the degree-shift identity and step 2.1 the equivalent $h^{0} - h^{1}$ identity, both in $Z$ because they are rearrangements of identities between finite alternating sums of finite dimensions and the integer $\deg_{k} (D)$ ; step 3.1 records the equivalence. The Axiom of Choice is used only through the suppliers recorded in [F6], namely the decomposition lemma [F2], the finiteness and coherence supplier [F3] and the flagged dictionary [F5]; in particular no ordering of the divisor support is chosen in a way that needs any choice principle, the listings being finite and fixed by the divisor, and the ordering-independence clause of [F2] holds for every ordering.

Depends on

Used by

Dependency tree · two levels

85 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)