Vanishing of the cohomology of line bundles on a curve above degree one

On a Noetherian integral scheme whose codimension-one points are closed and have discrete valuation rings as local rings, the sheaf 𝒪_X(D) of a Weil divisor has the flasque resolution 0 ⟶ 𝒪_X(D) ⟶ 𝒦_X ⟶ 𝒦_X / 𝒪_X(D) ⟶ 0 of TauCeti/AlgebraicGeometry/WeilDivisor/Scheme/PrincipalParts/Basic.lean. Flasque sheaves are acyclic, so the long exact cohomology sequence gives Hⁱ(X, 𝒪_X(D)) = 0 for i ≥ 2. Since every line bundle on a curve is isomorphic to some 𝒪_X(D), the same holds for every line bundle.

Main declarations

• SchemeWeilDivisor.subsingleton_cohomology_sheaf_add_two: Hⁿ⁺²(X, 𝒪_X(D)) = 0 when the codimension-one points are closed;
• SchemeWeilDivisor.subsingleton_cohomology_add_two_of_invertibleSheaf: Hⁿ⁺²(X, L) = 0 for every invertible sheaf L on a Noetherian integral scheme of dimension at most one whose codimension-one local rings are discrete valuation rings.

In degree one the same resolution identifies H¹(X, 𝒪_X(D)) with the cokernel of H⁰(X, 𝒦_X) ⟶ H⁰(X, 𝒦_X / 𝒪_X(D)), by Scheme.Modules.cohomologyOneLinearEquivOfIsFlasque applied to SchemeWeilDivisor.principalPartsShortComplex_shortExact.

The acyclicity of flasque sheaves is Scheme.Modules.subsingleton_cohomology_succ_of_isFlasque, the long exact sequence is TauCeti/AlgebraicGeometry/Cohomology/LongExactSequence.lean, and the comparison of line bundles with divisor sheaves is SchemeWeilDivisor.exists_nonempty_iso_sheaf.

References

• R. Hartshorne, Algebraic Geometry, III, Proposition 2.5 (flasque sheaves are acyclic) and Theorem 2.7 (Grothendieck vanishing, of which the statements here are a special case).
• J.-P. Serre, Algebraic Groups and Class Fields, Chapter II, §5.

theorem TauCeti.AlgebraicGeometry.SchemeWeilDivisor.subsingleton_cohomology_sheaf_add_two {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsIntegral X] [AlgebraicGeometry.IsNoetherian X] [∀ (x : CodimensionOnePoint X), IsDiscreteValuationRing ↑(X.presheaf.stalk ↑x)] (hclosed : ∀ (x : CodimensionOnePoint X), IsClosed {↑x}) (D : SchemeWeilDivisor X) (n : ℕ) : Subsingleton (Scheme.Modules.Cohomology D.sheaf (n + 2))

The sheaf of a Weil divisor has no cohomology above degree one on a Noetherian integral scheme whose codimension-one points are closed and have discrete valuation rings as local rings: the flasque resolution 0 ⟶ 𝒪_X(D) ⟶ 𝒦_X ⟶ 𝒦_X / 𝒪_X(D) ⟶ 0 has length one.

theorem TauCeti.AlgebraicGeometry.SchemeWeilDivisor.subsingleton_cohomology_add_two_of_invertibleSheaf {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsIntegral X] [AlgebraicGeometry.IsNoetherian X] [∀ (x : CodimensionOnePoint X), IsDiscreteValuationRing ↑(X.presheaf.stalk ↑x)] (hX : ∀ (y : ↥X), Order.coheight y ≤ 1) (L : InvertibleSheaf X) (n : ℕ) : Subsingleton (Scheme.Modules.Cohomology L.obj (n + 2))

Line bundles on a curve have no cohomology above degree one. On a Noetherian integral scheme of dimension at most one whose codimension-one local rings are discrete valuation rings, Hⁱ(X, L) = 0 for every invertible sheaf L and every i ≥ 2.