These are the geometric definitions used in Yamaguchi, arXiv:2610.02948v1: reduced prime components of an effective Cartier divisor, their generic points, horizontality over Spec ℤ, and geometric integrality after finite etale covers. Schemes and geometric point fields are in universe zero.
The ambient scheme is not assumed globally integral, normal, reduced, or of finite type. The generic-fiber condition is imposed on individual reduced boundary components. The maximal prosolvable quotient uses the existing closed residual core.
The prime components of the divisor (their schemes have reduced structure).
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
namespace Yamaguchi26SolvableCoveringsabbrev Component {X : Scheme.{0}} (D : X.EffectiveCartierDivisor)The generic point of a prime component, regarded as a point of X.
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
namespace Yamaguchi26SolvableCoveringsdef componentPoint {X : Scheme.{0}} (D : X.EffectiveCartierDivisor)
(E : Component D) : XThe canonical arithmetic structure morphism; it is not extra input data.
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
namespace Yamaguchi26SolvableCoveringsdef toIntegers (X : Scheme.{0}) : X ⟶ Spec (CommRingCat.of ℤ)Every prime component of D dominates Spec ℤ.
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
namespace Yamaguchi26SolvableCoveringsdef IsHorizontal {X : Scheme.{0}} (D : X.EffectiveCartierDivisor) : PropGeometric integrality of each reduced boundary component's generic fiber over ℚ, rather than geometric integrality of the entire arithmetic morphism.
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
namespace Yamaguchi26SolvableCoveringsdef HasGeometricallyIntegralGenericComponents {X : Scheme.{0}}
(I : X.IdealSheafData) : PropThe geometric-integrality hypothesis after every connected finite étale cover, including covers which are not Galois.
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
namespace Yamaguchi26SolvableCoveringsdef HasGeometricallyIntegralGenericComponentsAfterCovers {X : Scheme.{0}}
(D : X.EffectiveCartierDivisor) : PropThe maximal prosolvable quotient, using the existing closed residual core.
Surrounding namespaces, section variables, and local commands at this declaration.
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
namespace Yamaguchi26SolvableCoveringsabbrev Prosolvable (G : Type*) [Group G] [TopologicalSpace G]
[IsTopologicalGroup G]