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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 | import Mathlib
namespace OAI
section
noncomputable section
open Function Module
open scoped Classical
namespace CriticalPercolation.LinearGrowth
lemma toModuleEnd_surjective_of_simple {K A V : Type*} [Field K] [IsAlgClosed K]
[Ring A] [Algebra K A] [AddCommGroup V] [Module K V] [Module A V]
[IsScalarTower K A V] [IsSimpleModule A V] [FiniteDimensional K V] :
Surjective (Module.toModuleEnd K (S:=A) V) := by
let E := Module.End A V
have he : Surjective (algebraMap K E) :=
(IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed K).2
let : Module.Finite E V := Module.Finite.of_restrictScalars_finite K E V
intro T
let T' : Module.End E V := {
toFun := T
map_add' := T.map_add
map_smul' := by
intro e x
obtain ⟨c,rfl⟩ := he e
change T (c • x)=c • T x
exact T.map_smul c x }
obtain ⟨a,ha⟩ := Module.Finite.toModuleEnd_moduleEnd_surjective (R:=A) (M:=V) T'
refine ⟨a,?_⟩
ext x
exact congrArg (fun f : Module.End E V => f x) ha
lemma subalgebra_eq_top_of_simple {K V : Type*} [Field K] [IsAlgClosed K]
[AddCommGroup V] [Module K V] [FiniteDimensional K V]
(A : Subalgebra K (Module.End K V))
[IsSimpleModule A V] : A=⊤ := by
apply top_unique
intro T _
obtain ⟨a,ha⟩ := toModuleEnd_surjective_of_simple (K:=K) (A:=A) (V:=V) T
have he : (a : Module.End K V)=T := by
ext v
exact congrArg (fun f : Module.End K V => f v) ha
exact he ▸ a.property
end CriticalPercolation.LinearGrowth
end
end
end OAI
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