ModuleGeometricGenus.lean
Nyll.AlgebraicCurves.Geometry.Curves.GeometricGenus
2 declarations indexed from sourceView source ↗ Module documentation
Geometric genus and structure-sheaf cohomology over an arbitrary field
The genus is defined by the actual function field of the base change to an algebraic closure. It needs neither a rational point nor a chosen affine cover. For a smooth proper geometrically integral curve it equals the dimension of structure-sheaf first cohomology on every two-affine cover of the original curve.
Declarations
AlgebraicGeometry.SmoothProperCurve.geometricGenus
DocumentationThe genus of the geometric function field, using its actual repartition cohomology.
Source context 5 commands
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits
open scoped TensorProduct
namespace AlgebraicGeometry.SmoothProperCurve
AlgebraicGeometry.SmoothProperCurve.finrank_H1_eq_geometricGenus
DocumentationScalar extension preserves the dimension of the actual Cech cokernel, identifying the point-free geometric genus with first cohomology over the original field.
Source context 5 commands
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
universe u
open CategoryTheory CategoryTheory.Limits
open scoped TensorProduct
namespace AlgebraicGeometry.SmoothProperCurve