File: Mathlib/RingTheory/Kaehler/Polynomial.lean

full_name: KaehlerDifferential.mvPolynomialEquiv
code: /-- The relative differential module of a polynomial algebra `R[σ]` is the free module generated by
`{ dx | x ∈ σ }`. Also see `KaehlerDifferential.mvPolynomialBasis`. -/
def KaehlerDifferential.mvPolynomialEquiv (σ : Type*) :
    Ω[MvPolynomial σ R⁄R] ≃ₗ[MvPolynomial σ R] σ →₀ MvPolynomial σ R 
goal: Ω[MvPolynomial σ R⁄R] ≃ₗ[MvPolynomial σ R] σ →₀ MvPolynomial σ R

full_name: KaehlerDifferential.mvPolynomialBasis
code: /-- `{ dx | x ∈ σ }` forms a basis of the relative differential module
of a polynomial algebra `R[σ]`. -/
def KaehlerDifferential.mvPolynomialBasis (σ) :
    Basis σ (MvPolynomial σ R) (Ω[MvPolynomial σ R⁄R]) 
goal: Basis σ (MvPolynomial σ R) (Ω[MvPolynomial σ R⁄R])

full_name: KaehlerDifferential.mvPolynomialBasis_repr_comp_D
code: lemma KaehlerDifferential.mvPolynomialBasis_repr_comp_D (σ) :
    (mvPolynomialBasis R σ).repr.toLinearMap.compDer (D _ _) =
      MvPolynomial.mkDerivation _ (Finsupp.single · 1) 
goal: (↑(KaehlerDifferential.mvPolynomialBasis R σ).repr).compDer (KaehlerDifferential.D R (MvPolynomial σ R)) =
  MvPolynomial.mkDerivation R fun x => Finsupp.single x 1

full_name: KaehlerDifferential.mvPolynomialBasis_repr_D
code: lemma KaehlerDifferential.mvPolynomialBasis_repr_D (σ) (x) :
    (mvPolynomialBasis R σ).repr (D _ _ x) =
      MvPolynomial.mkDerivation R (Finsupp.single · (1 : MvPolynomial σ R)) x 
goal: (KaehlerDifferential.mvPolynomialBasis R σ).repr ((KaehlerDifferential.D R (MvPolynomial σ R)) x) =
  (MvPolynomial.mkDerivation R fun x => Finsupp.single x 1) x

full_name: KaehlerDifferential.mvPolynomialBasis_repr_D_X
code: @[simp]
lemma KaehlerDifferential.mvPolynomialBasis_repr_D_X (σ) (i) :
    (mvPolynomialBasis R σ).repr (D _ _ (.X i)) = Finsupp.single i 1 
goal: (KaehlerDifferential.mvPolynomialBasis R σ).repr ((KaehlerDifferential.D R (MvPolynomial σ R)) (MvPolynomial.X i)) =
  Finsupp.single i 1

full_name: KaehlerDifferential.mvPolynomialBasis_repr_apply
code: @[simp]
lemma KaehlerDifferential.mvPolynomialBasis_repr_apply (σ) (x) (i) :
    (mvPolynomialBasis R σ).repr (D _ _ x) i = MvPolynomial.pderiv i x 
goal: ((KaehlerDifferential.mvPolynomialBasis R σ).repr ((KaehlerDifferential.D R (MvPolynomial σ R)) x)) i =
  (MvPolynomial.pderiv i) x

full_name: KaehlerDifferential.mvPolynomialBasis_repr_symm_single
code: lemma KaehlerDifferential.mvPolynomialBasis_repr_symm_single (σ) (i) (x) :
    (mvPolynomialBasis R σ).repr.symm (Finsupp.single i x) = x • D R (MvPolynomial σ R) (.X i) 
goal: (KaehlerDifferential.mvPolynomialBasis R σ).repr.symm (Finsupp.single i x) =
  x • (KaehlerDifferential.D R (MvPolynomial σ R)) (MvPolynomial.X i)

full_name: KaehlerDifferential.mvPolynomialBasis_apply
code: @[simp]
lemma KaehlerDifferential.mvPolynomialBasis_apply (σ) (i) :
    mvPolynomialBasis R σ i = D R (MvPolynomial σ R) (.X i) 
goal: (KaehlerDifferential.mvPolynomialBasis R σ) i = (KaehlerDifferential.D R (MvPolynomial σ R)) (MvPolynomial.X i)

full_name: instFreeMvPolynomialKaehlerDifferential
code: instance (σ) : Module.Free (MvPolynomial σ R) (Ω[MvPolynomial σ R⁄R]) 
goal: Module.Free (MvPolynomial σ R) (Ω[MvPolynomial σ R⁄R])

full_name: KaehlerDifferential.polynomial_D_apply
code: lemma KaehlerDifferential.polynomial_D_apply (P : R[X]) :
    D R R[X] P = derivative P • D R R[X] X 
goal: (KaehlerDifferential.D R (Polynomial R)) P =
  Polynomial.derivative P • (KaehlerDifferential.D R (Polynomial R)) Polynomial.X

full_name: KaehlerDifferential.polynomialEquiv
code: /-- The relative differential module of the univariate polynomial algebra `R[X]` is isomorphic to
  `R[X]` as an `R[X]`-module. -/
def KaehlerDifferential.polynomialEquiv : Ω[R[X]⁄R] ≃ₗ[R[X]] R[X] 
goal: Ω[Polynomial R⁄R] ≃ₗ[Polynomial R] Polynomial R

full_name: KaehlerDifferential.polynomialEquiv_comp_D
code: lemma KaehlerDifferential.polynomialEquiv_comp_D :
    (polynomialEquiv R).compDer (D R R[X]) = derivative' 
goal: (KaehlerDifferential.polynomialEquiv R).compDer (KaehlerDifferential.D R (Polynomial R)) = Polynomial.derivative'

full_name: KaehlerDifferential.polynomialEquiv_D
code: @[simp]
lemma KaehlerDifferential.polynomialEquiv_D (P) :
    polynomialEquiv R (D R R[X] P) = derivative P 
goal: (KaehlerDifferential.polynomialEquiv R) ((KaehlerDifferential.D R (Polynomial R)) P) = Polynomial.derivative P

full_name: KaehlerDifferential.polynomialEquiv_symm
code: @[simp]
lemma KaehlerDifferential.polynomialEquiv_symm (P) :
    (polynomialEquiv R).symm P = P • D R R[X] X 
goal: (KaehlerDifferential.polynomialEquiv R).symm P = P • (KaehlerDifferential.D R (Polynomial R)) Polynomial.X