The module of kaehler differentials

Main results

• KaehlerDifferential: The module of kaehler differentials. For an R-algebra S, we provide the notation Ω[S⁄R] for KaehlerDifferential R S. Note that the slash is \textfractionsolidus.
• KaehlerDifferential.D: The derivation into the module of kaehler differentials.
• KaehlerDifferential.span_range_derivation: The image of D spans Ω[S⁄R] as an S-module.
• KaehlerDifferential.linearMapEquivDerivation: The isomorphism Hom_R(Ω[S⁄R], M) ≃ₗ[S] Der_R(S, M).
• KaehlerDifferential.quotKerTotalEquiv: An alternative description of Ω[S⁄R] as S copies of S with kernel (KaehlerDifferential.kerTotal) generated by the relations:
  1. dx + dy = d(x + y)
  2. x dy + y dx = d(x * y)
  3. dr = 0 for r ∈ R
• KaehlerDifferential.map: Given a map between the arrows R → A and S → B, we have an A-linear map Ω[A⁄R] → Ω[B⁄S].

Future project

• Define the IsKaehlerDifferential predicate.

@[inline, reducible]

abbrev KaehlerDifferential.ideal (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Ideal (TensorProduct R S S)

The kernel of the multiplication map S ⊗[R] S →ₐ[R] S.

theorem KaehlerDifferential.one_smul_sub_smul_one_mem_ideal (R : Type u) {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (a : S) : 1 ⊗ₜ[R] a - a ⊗ₜ[R] 1 ∈ KaehlerDifferential.ideal R S

def Derivation.tensorProductTo {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) : TensorProduct R S S →ₗ[S] M

For a R-derivation S → M, this is the map S ⊗[R] S →ₗ[S] M sending s ⊗ₜ t ↦ s • D t.

theorem Derivation.tensorProductTo_tmul {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) (s : S) (t : S) : ↑(Derivation.tensorProductTo D) (s ⊗ₜ[R] t) = s • ↑D t

theorem Derivation.tensorProductTo_mul {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) (x : TensorProduct R S S) (y : TensorProduct R S S) : ↑(Derivation.tensorProductTo D) (x * y) = ↑(Algebra.TensorProduct.lmul' R) x • ↑(Derivation.tensorProductTo D) y + ↑(Algebra.TensorProduct.lmul' R) y • ↑(Derivation.tensorProductTo D) x

theorem KaehlerDifferential.submodule_span_range_eq_ideal (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Submodule.span S (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) = Submodule.restrictScalars S (KaehlerDifferential.ideal R S)

The kernel of S ⊗[R] S →ₐ[R] S is generated by 1 ⊗ s - s ⊗ 1 as a S-module.

theorem KaehlerDifferential.span_range_eq_ideal (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Ideal.span (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) = KaehlerDifferential.ideal R S

def KaehlerDifferential (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Type v

The module of Kähler differentials (Kahler differentials, Kaehler differentials). This is implemented as I / I ^ 2 with I the kernel of the multiplication map S ⊗[R] S →ₐ[R] S. To view elements as a linear combination of the form s • D s', use KaehlerDifferential.tensorProductTo_surjective and Derivation.tensorProductTo_tmul.

We also provide the notation Ω[S⁄R] for KaehlerDifferential R S. Note that the slash is \textfractionsolidus.

instance instAddCommGroupKaehlerDifferential (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : AddCommGroup (Ω[S⁄R])

instance KaehlerDifferential.module (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Module (TensorProduct R S S) (Ω[S⁄R])

def «termΩ[_⁄_]» : Lean.ParserDescr

The module of Kähler differentials (Kahler differentials, Kaehler differentials). This is implemented as I / I ^ 2 with I the kernel of the multiplication map S ⊗[R] S →ₐ[R] S. To view elements as a linear combination of the form s • D s', use KaehlerDifferential.tensorProductTo_surjective and Derivation.tensorProductTo_tmul.

We also provide the notation Ω[S⁄R] for KaehlerDifferential R S. Note that the slash is \textfractionsolidus.

instance instNonemptyKaehlerDifferential (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Nonempty (Ω[S⁄R])

instance KaehlerDifferential.module' (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] {R' : Type u_2} [CommRing R'] [Algebra R' S] [SMulCommClass R R' S] : Module R' (Ω[S⁄R])

instance instIsScalarTowerTensorProductToCommSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToRingToModuleToSemiringToCommSemiringKaehlerDifferentialLeftHasSMulToMonoidToMonoidWithZeroToDistribMulActionToModuleTo_smulCommClassToSMulToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidInstAddCommGroupKaehlerDifferentialToSMulZeroClassToZeroToCommMonoidWithZeroInstCommRingToSMulWithZeroInstSemiringToMulActionWithZeroToAddCommMonoidModuleModule'Id (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : IsScalarTower S (TensorProduct R S S) (Ω[S⁄R])

instance KaehlerDifferential.isScalarTower_of_tower (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] {R₁ : Type u_2} {R₂ : Type u_3} [CommRing R₁] [CommRing R₂] [Algebra R₁ S] [Algebra R₂ S] [SMul R₁ R₂] [SMulCommClass R R₁ S] [SMulCommClass R R₂ S] [IsScalarTower R₁ R₂ S] : IsScalarTower R₁ R₂ (Ω[S⁄R])

instance KaehlerDifferential.isScalarTower' (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : IsScalarTower R (TensorProduct R S S) (Ω[S⁄R])

def KaehlerDifferential.fromIdeal (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : { x // x ∈ KaehlerDifferential.ideal R S } →ₗ[TensorProduct R S S] Ω[S⁄R]

The quotient map I → Ω[S⁄R] with I being the kernel of S ⊗[R] S → S.

def KaehlerDifferential.DLinearMap (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : S →ₗ[R] Ω[S⁄R]

(Implementation) The underlying linear map of the derivation into Ω[S⁄R].

theorem KaehlerDifferential.DLinearMap_apply (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (s : S) : ↑(KaehlerDifferential.DLinearMap R S) s = ↑(Ideal.toCotangent (KaehlerDifferential.ideal R S)) { val := 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1, property := (_ : 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1 ∈ KaehlerDifferential.ideal R S) }

def KaehlerDifferential.D (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Derivation R S (Ω[S⁄R])

The universal derivation into Ω[S⁄R].

theorem KaehlerDifferential.D_apply (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (s : S) : ↑(KaehlerDifferential.D R S) s = ↑(Ideal.toCotangent (KaehlerDifferential.ideal R S)) { val := 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1, property := (_ : 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1 ∈ KaehlerDifferential.ideal R S) }

theorem KaehlerDifferential.span_range_derivation (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Submodule.span S (Set.range ↑(KaehlerDifferential.D R S)) = ⊤

def Derivation.liftKaehlerDifferential {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) : Ω[S⁄R] →ₗ[S] M

The linear map from Ω[S⁄R], associated with a derivation.

theorem Derivation.liftKaehlerDifferential_apply {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) (x : { x // x ∈ KaehlerDifferential.ideal R S }) : ↑(Derivation.liftKaehlerDifferential D) (↑(Ideal.toCotangent (KaehlerDifferential.ideal R S)) x) = ↑(Derivation.tensorProductTo D) ↑x

theorem Derivation.liftKaehlerDifferential_comp {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D : Derivation R S M) : ↑(LinearMap.compDer (Derivation.liftKaehlerDifferential D)) (KaehlerDifferential.D R S) = D

@[simp]

theorem Derivation.liftKaehlerDifferential_comp_D {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (D' : Derivation R S M) (x : S) : ↑(Derivation.liftKaehlerDifferential D') (↑(KaehlerDifferential.D R S) x) = ↑D' x

theorem Derivation.liftKaehlerDifferential_unique {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] (f : Ω[S⁄R] →ₗ[S] M) (f' : Ω[S⁄R] →ₗ[S] M) (hf : ↑(LinearMap.compDer f) (KaehlerDifferential.D R S) = ↑(LinearMap.compDer f') (KaehlerDifferential.D R S)) : f = f'

theorem Derivation.liftKaehlerDifferential_D (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Derivation.liftKaehlerDifferential (KaehlerDifferential.D R S) = LinearMap.id

theorem KaehlerDifferential.D_tensorProductTo {R : Type u} {S : Type v} [CommRing R] [CommRing S] [Algebra R S] (x : { x // x ∈ KaehlerDifferential.ideal R S }) : ↑(Derivation.tensorProductTo (KaehlerDifferential.D R S)) ↑x = ↑(Ideal.toCotangent (KaehlerDifferential.ideal R S)) x

theorem KaehlerDifferential.tensorProductTo_surjective (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Function.Surjective ↑(Derivation.tensorProductTo (KaehlerDifferential.D R S))

def KaehlerDifferential.linearMapEquivDerivation (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] {M : Type u_1} [AddCommGroup M] [Module R M] [Module S M] [IsScalarTower R S M] : (Ω[S⁄R] →ₗ[S] M) ≃ₗ[S] Derivation R S M

The S-linear maps from Ω[S⁄R] to M are (S-linearly) equivalent to R-derivations from S to M.

def KaehlerDifferential.quotientCotangentIdealRingEquiv (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) ⧸ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S) ≃+* S

The quotient ring of S ⊗ S ⧸ J ^ 2 by Ω[S⁄R] is isomorphic to S.

def KaehlerDifferential.quotientCotangentIdeal (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : ((TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) ⧸ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S)) ≃ₐ[S] S

The quotient ring of S ⊗ S ⧸ J ^ 2 by Ω[S⁄R] is isomorphic to S as an S-algebra.

theorem KaehlerDifferential.End_equiv_aux (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (f : S →ₐ[R] TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) : AlgHom.comp (Ideal.Quotient.mkₐ R (Ideal.cotangentIdeal (KaehlerDifferential.ideal R S))) f = IsScalarTower.toAlgHom R S ((TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) ⧸ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S)) ↔ AlgHom.comp (AlgHom.kerSquareLift (Algebra.TensorProduct.lmul' R)) f = AlgHom.id R S

def smul_SSmod_SSmod (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : SMul (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2)

A shortcut instance to prevent timing out. Hopefully to be removed in the future.

def isScalarTower_S_right (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : IsScalarTower S (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2)

A shortcut instance to prevent timing out. Hopefully to be removed in the future.

def isScalarTower_R_right (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : IsScalarTower R (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2)

A shortcut instance to prevent timing out. Hopefully to be removed in the future.

def isScalarTower_SS_right (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : IsScalarTower (TensorProduct R S S) (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) (TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2)

A shortcut instance to prevent timing out. Hopefully to be removed in the future.

def instS (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Module S { x // x ∈ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S) }

A shortcut instance to prevent timing out. Hopefully to be removed in the future.

def instR (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Module R { x // x ∈ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S) }

A shortcut instance to prevent timing out. Hopefully to be removed in the future.

def instSS (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Module (TensorProduct R S S) { x // x ∈ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S) }

A shortcut instance to prevent timing out. Hopefully to be removed in the future.

noncomputable def KaehlerDifferential.endEquivDerivation' (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Derivation R S (Ω[S⁄R]) ≃ₗ[R] Derivation R S { x // x ∈ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S) }

Derivations into Ω[S⁄R] is equivalent to derivations into (KaehlerDifferential.ideal R S).cotangentIdeal.

def KaehlerDifferential.endEquivAuxEquiv (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : { f // AlgHom.comp (Ideal.Quotient.mkₐ R (Ideal.cotangentIdeal (KaehlerDifferential.ideal R S))) f = IsScalarTower.toAlgHom R S ((TensorProduct R S S ⧸ KaehlerDifferential.ideal R S ^ 2) ⧸ Ideal.cotangentIdeal (KaehlerDifferential.ideal R S)) } ≃ { f // AlgHom.comp (AlgHom.kerSquareLift (Algebra.TensorProduct.lmul' R)) f = AlgHom.id R S }

(Implementation) An Equiv version of KaehlerDifferential.End_equiv_aux. Used in KaehlerDifferential.endEquiv.

noncomputable def KaehlerDifferential.endEquiv (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Module.End S (Ω[S⁄R]) ≃ { f // AlgHom.comp (AlgHom.kerSquareLift (Algebra.TensorProduct.lmul' R)) f = AlgHom.id R S }

The endomorphisms of Ω[S⁄R] corresponds to sections of the surjection S ⊗[R] S ⧸ J ^ 2 →ₐ[R] S, with J being the kernel of the multiplication map S ⊗[R] S →ₐ[R] S.

noncomputable def KaehlerDifferential.kerTotal (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Submodule S (S →₀ S)

The S-submodule of S →₀ S (the direct sum of copies of S indexed by S) generated by the relations:

1. dx + dy = d(x + y)
2. x dy + y dx = d(x * y)
3. dr = 0 for r ∈ R where db is the unit in the copy of S with index b.

This is the kernel of the surjection Finsupp.total S Ω[S⁄R] S (KaehlerDifferential.D R S). See KaehlerDifferential.kerTotal_eq and KaehlerDifferential.total_surjective.

noncomputable def KaehlerDifferential.derivationQuotKerTotal (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Derivation R S ((S →₀ S) ⧸ KaehlerDifferential.kerTotal R S)

The (universal) derivation into (S →₀ S) ⧸ KaehlerDifferential.kerTotal R S.

theorem KaehlerDifferential.derivationQuotKerTotal_lift_comp_total (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : LinearMap.comp (Derivation.liftKaehlerDifferential (KaehlerDifferential.derivationQuotKerTotal R S)) (Finsupp.total S (Ω[S⁄R]) S ↑(KaehlerDifferential.D R S)) = Submodule.mkQ (KaehlerDifferential.kerTotal R S)

theorem KaehlerDifferential.kerTotal_eq (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : LinearMap.ker (Finsupp.total S (Ω[S⁄R]) S ↑(KaehlerDifferential.D R S)) = KaehlerDifferential.kerTotal R S

theorem KaehlerDifferential.total_surjective (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : Function.Surjective ↑(Finsupp.total S (Ω[S⁄R]) S ↑(KaehlerDifferential.D R S))

@[simp]

theorem KaehlerDifferential.quotKerTotalEquiv_symm_apply (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (a : Ω[S⁄R]) : ↑(LinearEquiv.symm (KaehlerDifferential.quotKerTotalEquiv R S)) a = ↑(Derivation.liftKaehlerDifferential (KaehlerDifferential.derivationQuotKerTotal R S)) a

@[simp]

theorem KaehlerDifferential.quotKerTotalEquiv_apply (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : ∀ (a : (S →₀ S) ⧸ Submodule.toAddSubgroup (KaehlerDifferential.kerTotal R S)), ↑(KaehlerDifferential.quotKerTotalEquiv R S) a = ↑(QuotientAddGroup.lift (Submodule.toAddSubgroup (KaehlerDifferential.kerTotal R S)) (LinearMap.toAddMonoidHom (Finsupp.total S (Ω[S⁄R]) S ↑(KaehlerDifferential.D R S))) (_ : KaehlerDifferential.kerTotal R S ≤ LinearMap.ker (Finsupp.total S (Ω[S⁄R]) S ↑(KaehlerDifferential.D R S)))) a

noncomputable def KaehlerDifferential.quotKerTotalEquiv (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : ((S →₀ S) ⧸ KaehlerDifferential.kerTotal R S) ≃ₗ[S] Ω[S⁄R]

Ω[S⁄R] is isomorphic to S copies of S with kernel KaehlerDifferential.kerTotal.

theorem KaehlerDifferential.quotKerTotalEquiv_symm_comp_D (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] : ↑(LinearMap.compDer ↑(LinearEquiv.symm (KaehlerDifferential.quotKerTotalEquiv R S))) (KaehlerDifferential.D R S) = KaehlerDifferential.derivationQuotKerTotal R S

theorem KaehlerDifferential.kerTotal_map (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra S B] [Algebra R B] [Algebra A B] [IsScalarTower R S B] [IsScalarTower R A B] (h : Function.Surjective ↑(algebraMap A B)) : Submodule.map (LinearMap.comp (Finsupp.mapRange.linearMap (Algebra.linearMap A B)) (Finsupp.lmapDomain A A ↑(algebraMap A B))) (KaehlerDifferential.kerTotal R A) ⊔ Submodule.span A (Set.range fun x => fun₀ | ↑(algebraMap S B) x => 1) = Submodule.restrictScalars A (KaehlerDifferential.kerTotal S B)

def Derivation.compAlgebraMap {R : Type u} [CommRing R] {M : Type u_1} [AddCommGroup M] [Module R M] (A : Type u_2) {B : Type u_3} [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] [Module A M] [Module B M] [IsScalarTower A B M] (d : Derivation R B M) : Derivation R A M

For a tower R → A → B and an R-derivation B → M, we may compose with A → B to obtain an R-derivation A → M.

def KaehlerDifferential.map (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] : Ω[A⁄R] →ₗ[A] Ω[B⁄S]

The map Ω[A⁄R] →ₗ[A] Ω[B⁄R] given a square A --→ B ↑ ↑ | | R --→ S

theorem KaehlerDifferential.map_compDer (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] : ↑(LinearMap.compDer (KaehlerDifferential.map R S A B)) (KaehlerDifferential.D R A) = Derivation.compAlgebraMap A (Derivation.restrictScalars R (KaehlerDifferential.D S B))

theorem KaehlerDifferential.map_D (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] (x : A) : ↑(KaehlerDifferential.map R S A B) (↑(KaehlerDifferential.D R A) x) = ↑(KaehlerDifferential.D S B) (↑(algebraMap A B) x)

theorem KaehlerDifferential.map_surjective_of_surjective (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [Algebra S B] [IsScalarTower R A B] [IsScalarTower R S B] [SMulCommClass S A B] (h : Function.Surjective ↑(algebraMap A B)) : Function.Surjective ↑(KaehlerDifferential.map R S A B)

noncomputable def KaehlerDifferential.mapBaseChange (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] : TensorProduct A B (Ω[A⁄R]) →ₗ[B] Ω[B⁄R]

The lift of the map Ω[A⁄R] →ₗ[A] Ω[B⁄R] to the base change along A → B. This is the first map in the exact sequence B ⊗[A] Ω[A⁄R] → Ω[B⁄R] → Ω[B⁄A] → 0.

@[simp]

theorem KaehlerDifferential.mapBaseChange_tmul (R : Type u) [CommRing R] (A : Type u_2) (B : Type u_3) [CommRing A] [CommRing B] [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] (x : B) (y : Ω[A⁄R]) : ↑(KaehlerDifferential.mapBaseChange R A B) (x ⊗ₜ[A] y) = x • ↑(KaehlerDifferential.map R R A B) y