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chore: split RingTheory/Kaehler (#13978)
Put materials about (Mv)Polynomial into Kaehler/Polynomial and the others into Kaehler/Basic
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| /- | ||
| Copyright (c) 2024 Andrew Yang. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Andrew Yang | ||
| -/ | ||
| import Mathlib.RingTheory.Kaehler.Basic | ||
| import Mathlib.Algebra.MvPolynomial.PDeriv | ||
| import Mathlib.Algebra.Polynomial.Derivation | ||
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| /-! | ||
| # The Kaehler differential module of polynomial algebras | ||
| -/ | ||
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| open scoped TensorProduct | ||
| open Algebra | ||
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| universe u v | ||
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| variable (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] | ||
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| suppress_compilation | ||
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| section MvPolynomial | ||
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| /-- 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 where | ||
| __ := (MvPolynomial.mkDerivation _ (Finsupp.single · 1)).liftKaehlerDifferential | ||
| invFun := Finsupp.total σ _ _ (fun x ↦ D _ _ (MvPolynomial.X x)) | ||
| right_inv := by | ||
| intro x | ||
| induction' x using Finsupp.induction_linear with _ _ _ _ a b | ||
| · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]; rw [map_zero, map_zero] | ||
| · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, map_add] at *; simp only [*] | ||
| · simp [LinearMap.map_smul, -map_smul] | ||
| left_inv := by | ||
| intro x | ||
| obtain ⟨x, rfl⟩ := total_surjective _ _ x | ||
| induction' x using Finsupp.induction_linear with _ _ _ _ a b | ||
| · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]; rw [map_zero, map_zero, map_zero] | ||
| · simp only [map_add, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom] at *; simp only [*] | ||
| · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Finsupp.total_single, | ||
| LinearMap.map_smul, Derivation.liftKaehlerDifferential_comp_D] | ||
| congr 1 | ||
| induction a using MvPolynomial.induction_on | ||
| · simp only [MvPolynomial.derivation_C, map_zero] | ||
| · simp only [map_add, *] | ||
| · simp [*] | ||
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| /-- `{ 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]) := | ||
| ⟨mvPolynomialEquiv R σ⟩ | ||
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| lemma KaehlerDifferential.mvPolynomialBasis_repr_comp_D (σ) : | ||
| (mvPolynomialBasis R σ).repr.toLinearMap.compDer (D _ _) = | ||
| MvPolynomial.mkDerivation _ (Finsupp.single · 1) := | ||
| Derivation.liftKaehlerDifferential_comp _ | ||
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| lemma KaehlerDifferential.mvPolynomialBasis_repr_D (σ) (x) : | ||
| (mvPolynomialBasis R σ).repr (D _ _ x) = | ||
| MvPolynomial.mkDerivation R (Finsupp.single · (1 : MvPolynomial σ R)) x := | ||
| Derivation.congr_fun (mvPolynomialBasis_repr_comp_D R σ) x | ||
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| @[simp] | ||
| lemma KaehlerDifferential.mvPolynomialBasis_repr_D_X (σ) (i) : | ||
| (mvPolynomialBasis R σ).repr (D _ _ (.X i)) = Finsupp.single i 1 := by | ||
| simp [mvPolynomialBasis_repr_D] | ||
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| @[simp] | ||
| lemma KaehlerDifferential.mvPolynomialBasis_repr_apply (σ) (x) (i) : | ||
| (mvPolynomialBasis R σ).repr (D _ _ x) i = MvPolynomial.pderiv i x := by | ||
| classical | ||
| suffices ((Finsupp.lapply i).comp | ||
| (mvPolynomialBasis R σ).repr.toLinearMap).compDer (D _ _) = MvPolynomial.pderiv i by | ||
| rw [← this]; rfl | ||
| apply MvPolynomial.derivation_ext | ||
| intro j | ||
| simp [Finsupp.single_apply, Pi.single_apply] | ||
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| set_option backward.isDefEq.lazyWhnfCore false in -- See https://github.com/leanprover-community/mathlib4/issues/12534 | ||
| 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) := by | ||
| apply (mvPolynomialBasis R σ).repr.injective; simp [LinearEquiv.map_smul, -map_smul] | ||
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| set_option backward.isDefEq.lazyWhnfCore false in -- See https://github.com/leanprover-community/mathlib4/issues/12534 | ||
| @[simp] | ||
| lemma KaehlerDifferential.mvPolynomialBasis_apply (σ) (i) : | ||
| mvPolynomialBasis R σ i = D R (MvPolynomial σ R) (.X i) := | ||
| (mvPolynomialBasis_repr_symm_single R σ i 1).trans (one_smul _ _) | ||
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| instance (σ) : Module.Free (MvPolynomial σ R) (Ω[MvPolynomial σ R⁄R]) := | ||
| .of_basis (KaehlerDifferential.mvPolynomialBasis R σ) | ||
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| end MvPolynomial | ||
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| section Polynomial | ||
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| open Polynomial | ||
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| lemma KaehlerDifferential.polynomial_D_apply (P : R[X]) : | ||
| D R R[X] P = derivative P • D R R[X] X := by | ||
| rw [← aeval_X_left_apply P, (D R R[X]).map_aeval, aeval_X_left_apply, aeval_X_left_apply] | ||
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| /-- 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] where | ||
| __ := derivative'.liftKaehlerDifferential | ||
| invFun := (Algebra.lsmul R R _).toLinearMap.flip (D R R[X] X) | ||
| left_inv := by | ||
| intro x | ||
| obtain ⟨x, rfl⟩ := total_surjective _ _ x | ||
| induction' x using Finsupp.induction_linear with x y hx hy x y | ||
| · simp | ||
| · simp only [map_add, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, LinearMap.flip_apply, | ||
| AlgHom.toLinearMap_apply, lsmul_coe] at *; simp only [*] | ||
| · simp [polynomial_D_apply _ x] | ||
| right_inv x := by simp | ||
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| lemma KaehlerDifferential.polynomialEquiv_comp_D : | ||
| (polynomialEquiv R).compDer (D R R[X]) = derivative' := | ||
| Derivation.liftKaehlerDifferential_comp _ | ||
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| @[simp] | ||
| lemma KaehlerDifferential.polynomialEquiv_D (P) : | ||
| polynomialEquiv R (D R R[X] P) = derivative P := | ||
| Derivation.congr_fun (polynomialEquiv_comp_D R) P | ||
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| @[simp] | ||
| lemma KaehlerDifferential.polynomialEquiv_symm (P) : | ||
| (polynomialEquiv R).symm P = P • D R R[X] X := rfl | ||
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| end Polynomial |
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