Datasets:

roseking
/

openAI-math-dataset

import Mathlib

namespace OAI

noncomputable section
open CategoryTheory AlgebraicGeometry
open scoped TensorProduct

noncomputable section
open scoped TensorProduct
namespace ReverseLogKodaira.DifferentialBaseChange

section Linear
variable (A B M N : Type*) [CommRing A] [CommRing B] [Algebra A B]
  [AddCommGroup M] [Module A M] [AddCommGroup N] [Module A N] [Module B N]
  [IsScalarTower A B N]

def alternatingRestriction (n : ℕ) : N [⋀^Fin n]→ₗ[A] (⋀[B]^n N) where
  toMultilinearMap := (exteriorPower.ιMulti B n).toMultilinearMap.restrictScalars A
  map_eq_zero_of_eq' := (exteriorPower.ιMulti B n).map_eq_zero_of_eq

def exteriorScalarMap (f : M →ₗ[A] N) (n : ℕ) : (⋀[A]^n M) →ₗ[A] (⋀[B]^n N) :=
  exteriorPower.alternatingMapLinearEquiv
    ((alternatingRestriction A B N n).compLinearMap f)

def tensorScalarMap (f : M →ₗ[A] N) (m : ℕ) : (⨂[A]^m M) →ₗ[A] (⨂[B]^m N) :=
  PiTensorProduct.lift (((PiTensorProduct.tprod B).restrictScalars A).compLinearMap
    fun _ : Fin m => f)

end Linear

section Differentials
variable (R S A B : Type*) [CommRing R] [CommRing S] [CommRing A] [CommRing B]
  [Algebra R S] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B]
  [IsScalarTower R A B] [IsScalarTower R S B]

abbrev Canonical (n : ℕ) := ⋀[A]^n (KaehlerDifferential R A)
abbrev Pluricanonical (n m : ℕ) := ⨂[A]^m (Canonical R A n)

def exteriorDifferentialMap (n : ℕ) : Canonical R A n →ₗ[A] Canonical S B n :=
  exteriorScalarMap A B _ _ (KaehlerDifferential.map R S A B) n

def pluricanonicalMap (n m : ℕ) : Pluricanonical R A n m →ₗ[A] Pluricanonical S B n m :=
  tensorScalarMap A B _ _ (exteriorDifferentialMap R S A B n) m

end Differentials

section LinearProofs
variable {A B M N : Type*} [CommRing A] [CommRing B] [Algebra A B]
  [AddCommGroup M] [Module A M] [AddCommGroup N] [Module A N] [Module B N]
  [IsScalarTower A B N]

lemma lift_bijective_of_basis {I : Type*} (b : Module.Basis I A M)
    (c : Module.Basis I B N) (f : M →ₗ[A] N) (hf : ∀ i, f (b i) = c i) :
    Function.Bijective (f.liftBaseChange B) := by
  have he : f.liftBaseChange B = ((b.baseChange B).equiv c (Equiv.refl I)).toLinearMap := by
    apply (b.baseChange B).ext
    intro i
    rw [LinearEquiv.coe_toLinearMap, Module.Basis.equiv_apply]
    simp [Module.Basis.baseChange_apply, hf]
  rw [he]
  exact ((b.baseChange B).equiv c (Equiv.refl I)).bijective

@[simp] lemma exteriorScalarMap_wedge (f : M →ₗ[A] N) (n : ℕ) (v : Fin n → M) :
    exteriorScalarMap A B M N f n (exteriorPower.ιMulti A n v) =
      exteriorPower.ιMulti B n (f ∘ v) := by
  simp only [exteriorScalarMap, exteriorPower.alternatingMapLinearEquiv_apply_ιMulti,
    AlternatingMap.compLinearMap_apply]
  rfl

lemma exterior_bijective_of_basis {I : Type*} [LinearOrder I]
    (b : Module.Basis I A M) (c : Module.Basis I B N)
    (f : M →ₗ[A] N) (hf : ∀ i, f (b i) = c i) (n : ℕ) :
    Function.Bijective ((exteriorScalarMap A B M N f n).liftBaseChange B) := by
  apply lift_bijective_of_basis (b.exteriorPower n) (c.exteriorPower n)
  intro i
  simp only [exteriorPower.coe_basis, exteriorPower.ιMulti_family, exteriorScalarMap_wedge]
  congr 1
  funext j
  exact hf _

lemma tensor_bijective_of_basis {I : Type*}
    (b : Module.Basis I A M) (c : Module.Basis I B N)
    (f : M →ₗ[A] N) (hf : ∀ i, f (b i) = c i) (m : ℕ) :
    Function.Bijective ((tensorScalarMap A B M N f m).liftBaseChange B) := by
  classical
  apply lift_bijective_of_basis (Basis.piTensorProduct fun _ : Fin m => b)
    (Basis.piTensorProduct fun _ : Fin m => c)
  intro i
  simp only [Basis.piTensorProduct_apply, tensorScalarMap, PiTensorProduct.lift.tprod,
    MultilinearMap.compLinearMap_apply]
  change PiTensorProduct.tprod B (fun j => f (b (i j))) = PiTensorProduct.tprod B (fun j => c (i j))
  congr 1
  funext j
  exact hf _

end LinearProofs

section DifferentialProofs
variable (R S A B : Type*) [CommRing R] [CommRing S] [CommRing A] [CommRing B]
  [Algebra R S] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B]
  [IsScalarTower R A B] [IsScalarTower R S B] [Algebra.IsPushout R S A B]

lemma tensorKaehlerEquiv_tmul (b : B) (x : KaehlerDifferential R A) :
    KaehlerDifferential.tensorKaehlerEquiv R S A B (b ⊗ₜ[A] x) =
      b • KaehlerDifferential.map R S A B x := by
  have he : ((KaehlerDifferential.tensorKaehlerEquiv R S A B).toLinearMap.restrictScalars A).comp
      (TensorProduct.mk A B (KaehlerDifferential R A) b) =
      b • KaehlerDifferential.map R S A B := by
    apply LinearMap.ext_on (KaehlerDifferential.span_range_derivation R A)
    rintro _ ⟨a, rfl⟩
    simp
  exact LinearMap.congr_fun he x

 

theorem smooth_pluricanonical_baseChange
    [Algebra.IsStandardSmooth R A] (n m : ℕ) :
    Function.Bijective ((pluricanonicalMap R S A B n m).liftBaseChange B) := by
  classical
  let b := Module.Free.chooseBasis A (KaehlerDifferential R A)
  let : LinearOrder (Module.Free.ChooseBasisIndex A (KaehlerDifferential R A)) :=
    linearOrderOfSTO WellOrderingRel
  let c := (b.baseChange B).map (KaehlerDifferential.tensorKaehlerEquiv R S A B)
  have hc (i) : KaehlerDifferential.map R S A B (b i) = c i := by
    simp [c, Module.Basis.baseChange_apply, tensorKaehlerEquiv_tmul]
  apply tensor_bijective_of_basis (b.exteriorPower n) (c.exteriorPower n)
  intro i
  simp only [exteriorDifferentialMap, exteriorPower.coe_basis, exteriorPower.ιMulti_family,
    exteriorScalarMap_wedge]
  congr 1
  funext j
  exact hc _

end DifferentialProofs

end ReverseLogKodaira.DifferentialBaseChange

end
end

end OAI

	import Mathlib

	namespace OAI

	noncomputable section
	open CategoryTheory AlgebraicGeometry
	open scoped TensorProduct

	noncomputable section
	open scoped TensorProduct
	namespace ReverseLogKodaira.DifferentialBaseChange

	section Linear
	variable (A B M N : Type*) [CommRing A] [CommRing B] [Algebra A B]
	[AddCommGroup M] [Module A M] [AddCommGroup N] [Module A N] [Module B N]
	[IsScalarTower A B N]

	def alternatingRestriction (n : ℕ) : N [⋀^Fin n]→ₗ[A] (⋀[B]^n N) where
	toMultilinearMap := (exteriorPower.ιMulti B n).toMultilinearMap.restrictScalars A
	map_eq_zero_of_eq' := (exteriorPower.ιMulti B n).map_eq_zero_of_eq

	def exteriorScalarMap (f : M →ₗ[A] N) (n : ℕ) : (⋀[A]^n M) →ₗ[A] (⋀[B]^n N) :=
	exteriorPower.alternatingMapLinearEquiv
	((alternatingRestriction A B N n).compLinearMap f)

	def tensorScalarMap (f : M →ₗ[A] N) (m : ℕ) : (⨂[A]^m M) →ₗ[A] (⨂[B]^m N) :=
	PiTensorProduct.lift (((PiTensorProduct.tprod B).restrictScalars A).compLinearMap
	fun _ : Fin m => f)

	end Linear

	section Differentials
	variable (R S A B : Type*) [CommRing R] [CommRing S] [CommRing A] [CommRing B]
	[Algebra R S] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B]
	[IsScalarTower R A B] [IsScalarTower R S B]

	abbrev Canonical (n : ℕ) := ⋀[A]^n (KaehlerDifferential R A)
	abbrev Pluricanonical (n m : ℕ) := ⨂[A]^m (Canonical R A n)

	def exteriorDifferentialMap (n : ℕ) : Canonical R A n →ₗ[A] Canonical S B n :=
	exteriorScalarMap A B _ _ (KaehlerDifferential.map R S A B) n

	def pluricanonicalMap (n m : ℕ) : Pluricanonical R A n m →ₗ[A] Pluricanonical S B n m :=
	tensorScalarMap A B _ _ (exteriorDifferentialMap R S A B n) m

	end Differentials

	section LinearProofs
	variable {A B M N : Type*} [CommRing A] [CommRing B] [Algebra A B]
	[AddCommGroup M] [Module A M] [AddCommGroup N] [Module A N] [Module B N]
	[IsScalarTower A B N]

	lemma lift_bijective_of_basis {I : Type*} (b : Module.Basis I A M)
	(c : Module.Basis I B N) (f : M →ₗ[A] N) (hf : ∀ i, f (b i) = c i) :
	Function.Bijective (f.liftBaseChange B) := by
	have he : f.liftBaseChange B = ((b.baseChange B).equiv c (Equiv.refl I)).toLinearMap := by
	apply (b.baseChange B).ext
	intro i
	rw [LinearEquiv.coe_toLinearMap, Module.Basis.equiv_apply]
	simp [Module.Basis.baseChange_apply, hf]
	rw [he]
	exact ((b.baseChange B).equiv c (Equiv.refl I)).bijective

	@[simp] lemma exteriorScalarMap_wedge (f : M →ₗ[A] N) (n : ℕ) (v : Fin n → M) :
	exteriorScalarMap A B M N f n (exteriorPower.ιMulti A n v) =
	exteriorPower.ιMulti B n (f ∘ v) := by
	simp only [exteriorScalarMap, exteriorPower.alternatingMapLinearEquiv_apply_ιMulti,
	AlternatingMap.compLinearMap_apply]
	rfl

	lemma exterior_bijective_of_basis {I : Type*} [LinearOrder I]
	(b : Module.Basis I A M) (c : Module.Basis I B N)
	(f : M →ₗ[A] N) (hf : ∀ i, f (b i) = c i) (n : ℕ) :
	Function.Bijective ((exteriorScalarMap A B M N f n).liftBaseChange B) := by
	apply lift_bijective_of_basis (b.exteriorPower n) (c.exteriorPower n)
	intro i
	simp only [exteriorPower.coe_basis, exteriorPower.ιMulti_family, exteriorScalarMap_wedge]
	congr 1
	funext j
	exact hf _

	lemma tensor_bijective_of_basis {I : Type*}
	(b : Module.Basis I A M) (c : Module.Basis I B N)
	(f : M →ₗ[A] N) (hf : ∀ i, f (b i) = c i) (m : ℕ) :
	Function.Bijective ((tensorScalarMap A B M N f m).liftBaseChange B) := by
	classical
	apply lift_bijective_of_basis (Basis.piTensorProduct fun _ : Fin m => b)
	(Basis.piTensorProduct fun _ : Fin m => c)
	intro i
	simp only [Basis.piTensorProduct_apply, tensorScalarMap, PiTensorProduct.lift.tprod,
	MultilinearMap.compLinearMap_apply]
	change PiTensorProduct.tprod B (fun j => f (b (i j))) = PiTensorProduct.tprod B (fun j => c (i j))
	congr 1
	funext j
	exact hf _

	end LinearProofs

	section DifferentialProofs
	variable (R S A B : Type*) [CommRing R] [CommRing S] [CommRing A] [CommRing B]
	[Algebra R S] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B]
	[IsScalarTower R A B] [IsScalarTower R S B] [Algebra.IsPushout R S A B]

	lemma tensorKaehlerEquiv_tmul (b : B) (x : KaehlerDifferential R A) :
	KaehlerDifferential.tensorKaehlerEquiv R S A B (b ⊗ₜ[A] x) =
	b • KaehlerDifferential.map R S A B x := by
	have he : ((KaehlerDifferential.tensorKaehlerEquiv R S A B).toLinearMap.restrictScalars A).comp
	(TensorProduct.mk A B (KaehlerDifferential R A) b) =
	b • KaehlerDifferential.map R S A B := by
	apply LinearMap.ext_on (KaehlerDifferential.span_range_derivation R A)
	rintro _ ⟨a, rfl⟩
	simp
	exact LinearMap.congr_fun he x



	theorem smooth_pluricanonical_baseChange
	[Algebra.IsStandardSmooth R A] (n m : ℕ) :
	Function.Bijective ((pluricanonicalMap R S A B n m).liftBaseChange B) := by
	classical
	let b := Module.Free.chooseBasis A (KaehlerDifferential R A)
	let : LinearOrder (Module.Free.ChooseBasisIndex A (KaehlerDifferential R A)) :=
	linearOrderOfSTO WellOrderingRel
	let c := (b.baseChange B).map (KaehlerDifferential.tensorKaehlerEquiv R S A B)
	have hc (i) : KaehlerDifferential.map R S A B (b i) = c i := by
	simp [c, Module.Basis.baseChange_apply, tensorKaehlerEquiv_tmul]
	apply tensor_bijective_of_basis (b.exteriorPower n) (c.exteriorPower n)
	intro i
	simp only [exteriorDifferentialMap, exteriorPower.coe_basis, exteriorPower.ιMulti_family,
	exteriorScalarMap_wedge]
	congr 1
	funext j
	exact hc _

	end DifferentialProofs

	end ReverseLogKodaira.DifferentialBaseChange

	end
	end

	end OAI