Module

ActualFiniteTameDegreeDirected.lean

Nyll.EtaleFundamentalGroups.Inertia.ActualFiniteTameDegreeDirected
2 declarations indexed from sourceView source ↗

Directedness of the actual finite tame degrees

The literal range of the actual tame index is nonempty and directed by divisibility. Its upper bounds are supplied by finite Galois composita.

Declarations

EtaleFundamentalGroups.Inertia.actualFiniteTameDegreeSet_nonempty

A finite base stage supplies an actual tame degree.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

namespace EtaleFundamentalGroups.Inertia
open SchemeTheory
universe u v w
variable (A : Type u) (K : Type v) (L : Type w)
  [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
  [Field K] [Field L] [Algebra A K] [IsFractionRing A K]
  [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsGalois K L]
  (Q : Ideal (normalizationCommAlgCat A L).carrier) [Q.IsMaximal]
theorem actualFiniteTameDegreeSet_nonempty :
    (actualFiniteTameDegreeSet A K L Q).Nonempty
EtaleFundamentalGroups.Inertia.actualFiniteTameDegreeSet_directed

Actual finite Galois composita direct the literal tame-degree range.

Source context 4 commands

Surrounding namespaces, section variables, and local commands at this declaration.

namespace EtaleFundamentalGroups.Inertia
open SchemeTheory
universe u v w
variable (A : Type u) (K : Type v) (L : Type w)
  [CommRing A] [IsDomain A] [IsDiscreteValuationRing A]
  [Field K] [Field L] [Algebra A K] [IsFractionRing A K]
  [Algebra K L] [Algebra A L] [IsScalarTower A K L] [IsGalois K L]
  (Q : Ideal (normalizationCommAlgCat A L).carrier) [Q.IsMaximal]
theorem actualFiniteTameDegreeSet_directed :
    Directed (· ≤ ·) (id : actualFiniteTameDegreeSet A K L Q →
      actualFiniteTameDegreeSet A K L Q)