Riemann–Roch spaces on a proper curve #
For a Weil divisor D on an integral scheme whose codimension-one local rings are discrete
valuation rings, the global sections of the sheaf 𝒪_X(D) form the Riemann–Roch space
L(D) = {f ∈ K(X) | f = 0 or ord_y f ≥ -D(y) for every codimension-one point y}.
This file proves that L(D) is finite-dimensional over k when X is a proper curve over a
field k, and deduces that H⁰(X, L) is finite-dimensional for every line bundle L on such a
curve. This is the degree-zero half of the finiteness needed to read dim H⁰ - dim H¹ as the
Euler characteristic of a line bundle on a proper curve.
It also records what makes L(D) worth measuring: L(D) is nonzero exactly when the complete
linear system |D| of TauCeti.AlgebraicGeometry.WeilDivisor.LinearSystem.Basic is nonempty,
since a nonzero f ∈ L(D) is the same thing as an effective divisor D + div f in the class of
D. This part needs no properness and no curve hypothesis.
The argument is the classical one. Adding a codimension-one point y to D enlarges L(D) by at
most a copy of the residue field κ(y): if g has order D(y) + 1 at y, then f ↦ g f
sends L(D + y) into the local ring at y, and composing with the residue map gives a linear
map L(D + y) ⟶ κ(y) whose kernel is exactly L(D). Starting from L(0) = Γ(X, 𝒪_X), every
L(D) is reached by adding and removing points one at a time.
Main declarations #
SchemeWeilDivisor.residueMapandSchemeWeilDivisor.residueMap_eq_zero_iff: the residue linear map fromL(D + y)toκ(y)and the characterization of its kernel asL(D);SchemeWeilDivisor.fg_sections_add_ofPoint: ifL(D)is finitely generated overΓ(X, ⊤)andκ(y)is finite overΓ(X, ⊤), thenL(D + y)is finitely generated;SchemeWeilDivisor.fg_sections_of_le: over a Noetherian ring of global functions,L(D)is finitely generated as soon asL(E)is for someE ≥ D;SchemeWeilDivisor.fg_sections_top: on a curve whose ring of global functions is Noetherian and whose residue fields at codimension-one points are finite over it, everyL(D)is finitely generated;SchemeWeilDivisor.finiteDimensional_globalSections_sheafandSchemeWeilDivisor.finiteDimensional_cohomology_zero_sheaf: on a proper curve over a fieldk,Γ(X, 𝒪_X(D)) = H⁰(X, 𝒪_X(D))is finite-dimensional overk;InvertibleSheaf.finiteDimensional_cohomology_zero: on a proper curve overk,H⁰(X, L)is finite-dimensional for every line bundleL;SchemeWeilDivisor.nonempty_completeLinearSystem_iff_nontrivial_globalSections_sheaf: the complete linear system ofDis nonempty exactly whenΓ(X, 𝒪_X(D))is nonzero.
References #
- R. Hartshorne, Algebraic Geometry, Lemma IV.1.2 and Theorem III.5.2(a).
- W. Fulton, Algebraic Curves, Chapter 8, Proposition 2.
The whole of the integral scheme X is a nonempty open subset, so global sections of 𝒦_X
are rational functions.
The residue at y of g s, for a global section s of 𝒪_X(D + y) and a rational function
g of order D(y) + 1 at y, as a Γ(X, ⊤)-linear map to κ(y).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The kernel of residueMap D y hg0 hg consists exactly of the sections of 𝒪_X(D) inside
the sections of 𝒪_X(D + y).
Adding a point to a divisor preserves finite generation of global sections. If the global
sections of 𝒪_X(D) are finitely generated over a Noetherian ring Γ(X, ⊤), and the residue
field at y is a finite Γ(X, ⊤)-algebra, then the global sections of 𝒪_X(D + y) are finitely
generated: they are an extension of a submodule of κ(y) by those of 𝒪_X(D).
Over a Noetherian ring of global functions, the global sections of 𝒪_X(D) are finitely
generated as soon as those of 𝒪_X(E) are for some E ≥ D.
Global sections of 𝒪_X(D) are finitely generated. On a locally Noetherian integral scheme
of dimension at most one whose codimension-one local rings are discrete valuation rings, whose ring
Γ(X, ⊤) of global functions is Noetherian, and whose residue fields at codimension-one points
are finite Γ(X, ⊤)-algebras, the global sections of 𝒪_X(D) are a finitely generated
Γ(X, ⊤)-module for every Weil divisor D.
Riemann–Roch spaces are finite-dimensional. On a proper integral curve over a field k
whose codimension-one local rings are discrete valuation rings, the space Γ(X, 𝒪_X(D)) of global
sections of the sheaf of a Weil divisor D is finite-dimensional over k.
On a proper integral curve over a field k whose codimension-one local rings are discrete
valuation rings, H⁰(X, 𝒪_X(D)) is finite-dimensional over k for every Weil divisor D.
The complete linear system of D is nonempty exactly when 𝒪_X(D) has a nonzero global
section. On a Noetherian integral scheme whose codimension-one local rings are discrete
valuation rings, a global section of 𝒪_X(D) is a rational function f with D + div f ≥ 0, so
a nonzero one names an effective divisor linearly equivalent to D, and conversely.
H⁰ of a line bundle on a proper curve is finite-dimensional. On a proper integral curve
over a field k whose codimension-one local rings are discrete valuation rings, H⁰(X, L) is
finite-dimensional over k for every line bundle L: L is isomorphic to the sheaf 𝒪_X(D) of a
Weil divisor.