The genus and the Euler characteristic of the structure sheaf

For a scheme X over a field k with finite-dimensional H¹(X, 𝒪_X), the genus is g = dim_k H¹(X, 𝒪_X). For a smooth proper curve this is the usual genus.

The constant term 1 of that formula is dim_k H⁰(X, 𝒪_X). This file proves that it is 1 on an integral scheme that is universally closed (for instance proper) over k and has a k-rational point: the global functions are then the constants (appTop_bijective_of_section). Consequently the Euler characteristic of the structure sheaf is χ(𝒪_X) = 1 - g.

As in TauCeti.AlgebraicGeometry.LineBundle.Degree, the structure sheaf is written as the trivial line bundle (InvertibleSheaf.trivial X).obj, and Euler characteristics are the degree-2 truncations Scheme.Modules.eulerCharBelow k X M 2 = dim H⁰(X, M) - dim H¹(X, M). These are the Euler characteristics of a curve once cohomology above degree one is known to vanish. This file requires finite-dimensionality of H¹(X, 𝒪_X) when defining the genus; proving that finiteness for proper schemes is a separate prerequisite.

Main declarations

• AlgebraicGeometry.Scheme.genus k X, the genus dim_k H¹(X, 𝒪_X) of a scheme, and its defining formula AlgebraicGeometry.Scheme.genus_def;
• TauCeti.AlgebraicGeometry.finrank_cohomology_zero_trivial_eq_one: dim_k H⁰(X, 𝒪_X) = 1 for an integral scheme, universally closed over k, with a k-rational point;
• TauCeti.AlgebraicGeometry.eulerCharBelow_trivial_eq_one_sub_genus: for an integral scheme universally closed over k with a k-rational point, χ(𝒪_X) = 1 - g.

References

• R. Hartshorne, Algebraic Geometry, Exercise III.5.3 (the arithmetic genus).
• J.-P. Serre, Algebraic Groups and Class Fields, Chapter II, §3.

noncomputable def AlgebraicGeometry.Scheme.genus (k : Type u) [Field k] (X : Scheme) [X.Over (Spec ↧k)] [_fd : FiniteDimensional k (TauCeti.AlgebraicGeometry.Scheme.Modules.Cohomology (TauCeti.AlgebraicGeometry.InvertibleSheaf.trivial X).obj 1)] : ℕ

The genus dim_k H¹(X, 𝒪_X) of a scheme X over a field k, assuming this cohomology group is finite-dimensional. The finite-dimensionality instance _fd does not occur in the value — it guards the definition, so that finrank is never read as a genus 0 coming from its junk value on an infinite-dimensional space.

Equations

• AlgebraicGeometry.Scheme.genus k X = Module.finrank k (TauCeti.AlgebraicGeometry.Scheme.Modules.Cohomology (TauCeti.AlgebraicGeometry.InvertibleSheaf.trivial X).obj 1)

@[simp]

theorem AlgebraicGeometry.Scheme.genus_def (k : Type u) [Field k] (X : Scheme) [X.Over (Spec ↧k)] [FiniteDimensional k (TauCeti.AlgebraicGeometry.Scheme.Modules.Cohomology (TauCeti.AlgebraicGeometry.InvertibleSheaf.trivial X).obj 1)] : genus k X = Module.finrank k (TauCeti.AlgebraicGeometry.Scheme.Modules.Cohomology (TauCeti.AlgebraicGeometry.InvertibleSheaf.trivial X).obj 1)

The genus is the dimension of the first cohomology of the structure sheaf.

theorem TauCeti.AlgebraicGeometry.finrank_cohomology_zero_trivial_eq_one (k : Type u) [Field k] {X : AlgebraicGeometry.Scheme} [X.Over (AlgebraicGeometry.Spec ↧k)] [AlgebraicGeometry.IsIntegral X] [AlgebraicGeometry.UniversallyClosed (X ↘ AlgebraicGeometry.Spec ↧k)] {s : AlgebraicGeometry.Spec ↧k ⟶ X} (hs : CategoryTheory.CategoryStruct.comp s (X ↘ AlgebraicGeometry.Spec ↧k) = CategoryTheory.CategoryStruct.id (AlgebraicGeometry.Spec ↧k)) : Module.finrank k (Scheme.Modules.Cohomology (InvertibleSheaf.trivial X).obj 0) = 1

Only constant global functions. On an integral scheme that is universally closed over a field k and has a k-rational point, H⁰(X, 𝒪_X) is one-dimensional over k.

theorem TauCeti.AlgebraicGeometry.eulerCharBelow_trivial_eq_one_sub_genus (k : Type u) [Field k] {X : AlgebraicGeometry.Scheme} [X.Over (AlgebraicGeometry.Spec ↧k)] [AlgebraicGeometry.IsIntegral X] [FiniteDimensional k (Scheme.Modules.Cohomology (InvertibleSheaf.trivial X).obj 1)] [AlgebraicGeometry.UniversallyClosed (X ↘ AlgebraicGeometry.Spec ↧k)] {s : AlgebraicGeometry.Spec ↧k ⟶ X} (hs : CategoryTheory.CategoryStruct.comp s (X ↘ AlgebraicGeometry.Spec ↧k) = CategoryTheory.CategoryStruct.id (AlgebraicGeometry.Spec ↧k)) : AlgebraicGeometry.Scheme.Modules.eulerCharBelow k X (InvertibleSheaf.trivial X).obj 2 = 1 - ↑(AlgebraicGeometry.Scheme.genus k X)

The Euler characteristic of the structure sheaf. On an integral scheme universally closed over a field k with a k-rational point, χ(𝒪_X) = 1 - g, where χ is the degree-2 truncation dim H⁰ - dim H¹ and g is the genus.