Finiteness of the first cohomology of a proper curve

Let X be an integral scheme of dimension one, proper over a field k, whose codimension-one local rings are discrete valuation rings. Then H¹(X, 𝒪_X) is finite-dimensional over k.

This is the finiteness hypothesis under which the genus g = dim_k H¹(X, 𝒪_X) (AlgebraicGeometry.Scheme.genus) and the Riemann–Roch theorem (SchemeWeilDivisor.eulerCharBelow_sheaf_eq_relativeDegree_add_one_sub_genus) are stated, so both apply to every such curve.

Main declarations

• TauCeti.AlgebraicGeometry.finiteDimensional_cohomology_one_trivial_of_isProper: H¹(X, 𝒪_X) is finite-dimensional for a proper integral curve over k whose codimension-one local rings are discrete valuation rings.

References

• J.-P. Serre, Algebraic Groups and Class Fields, Chapter II, §5, Proposition 3.
• R. Hartshorne, Algebraic Geometry, Chapter III, Theorem 5.2.

theorem TauCeti.AlgebraicGeometry.finiteDimensional_cohomology_one_trivial_of_isProper (k : Type u) [Field k] {X : AlgebraicGeometry.Scheme} [X.Over (AlgebraicGeometry.Spec ↧k)] [AlgebraicGeometry.IsIntegral X] [∀ (y : CodimensionOnePoint X), IsDiscreteValuationRing ↑(X.presheaf.stalk ↑y)] [AlgebraicGeometry.IsProper (X ↘ AlgebraicGeometry.Spec ↧k)] (hX : topologicalKrullDim ↥X = 1) : FiniteDimensional k (Scheme.Modules.Cohomology (InvertibleSheaf.trivial X).obj 1)

The first cohomology of the structure sheaf of a proper curve is finite-dimensional. Let X be an integral scheme of dimension one, proper over a field k, whose codimension-one local rings are discrete valuation rings. Then H¹(X, 𝒪_X) is finite-dimensional over k.