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Mathlib.Algebra.Category.ModuleCat.Differentials.Basic

The differentials of a morphism in the category of commutative rings #

In this file, given a morphism f : A ⟶ B in the category CommRingCat, and M : ModuleCat B, we define the type M.Derivation f of derivations with values in M relative to f. We also construct the module of differentials CommRingCat.KaehlerDifferential f : ModuleCat B and the corresponding derivation.

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def ModuleCat.Derivation {A B : CommRingCat} (M : ModuleCat ↑B) (f : A ⟶ B) :
Type (max v u)

The type of derivations with values in a B-module M relative to a morphism f : A ⟶ B in the category CommRingCat.

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    def ModuleCat.Derivation.mk {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (d : ↑B → ↑M) (d_add : ∀ (b b' : ↑B), d (b + b') = d b + d b' := by simp) (d_mul : ∀ (b b' : ↑B), d (b * b') = b • d b' + b' • d b := by simp) (d_map : ∀ (a : ↑A), d (f.hom a) = 0 := by simp) :
    M.Derivation f

    Constructor for ModuleCat.Derivation.

    Equations
    • ModuleCat.Derivation.mk d d_add d_mul d_map = { toFun := d, map_add' := d_add, map_smul' := ⋯, map_one_eq_zero' := ⋯, leibniz' := d_mul }
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      def ModuleCat.Derivation.d {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (D : M.Derivation f) (b : ↑B) :
      ↑M

      The underlying map B → M of a derivation M.Derivation f when M : ModuleCat B and f : A ⟶ B is a morphism in CommRingCat.

      Equations
      • D.d b = ↑D b
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        @[simp]
        theorem ModuleCat.Derivation.d_add {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (D : M.Derivation f) (b b' : ↑B) :
        D.d (b + b') = D.d b + D.d b'
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        @[simp]
        theorem ModuleCat.Derivation.d_mul {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (D : M.Derivation f) (b b' : ↑B) :
        D.d (b * b') = b • D.d b' + b' • D.d b
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        @[simp]
        theorem ModuleCat.Derivation.d_map {A B : CommRingCat} {M : ModuleCat ↑B} {f : A ⟶ B} (D : M.Derivation f) (a : ↑A) :
        D.d (f.hom a) = 0
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        noncomputable def CommRingCat.KaehlerDifferential {A B : CommRingCat} (f : A ⟶ B) :
        ModuleCat ↑B

        The module of differentials of a morphism f : A ⟶ B in the category CommRingCat.

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          noncomputable def CommRingCat.KaehlerDifferential.D {A B : CommRingCat} (f : A ⟶ B) :
          (CommRingCat.KaehlerDifferential f).Derivation f

          The (universal) derivation in (KaehlerDifferential f).Derivation f when f : A ⟶ B is a morphism in the category CommRingCat.

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            @[reducible, inline]
            noncomputable abbrev CommRingCat.KaehlerDifferential.d {A B : CommRingCat} {f : A ⟶ B} (b : ↑B) :
            ↑(CommRingCat.KaehlerDifferential f)

            When f : A ⟶ B is a morphism in the category CommRingCat, this is the differential map B → KaehlerDifferential f.

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              theorem CommRingCat.KaehlerDifferential.ext {A B : CommRingCat} {f : A ⟶ B} {M : ModuleCat ↑B} {α β : CommRingCat.KaehlerDifferential f ⟶ M} (h : ∀ (b : ↑B), α.hom (CommRingCat.KaehlerDifferential.d b) = β.hom (CommRingCat.KaehlerDifferential.d b)) :
              α = β
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              noncomputable def CommRingCat.KaehlerDifferential.map {A B A' B' : CommRingCat} {f : A ⟶ B} {f' : A' ⟶ B'} {g : A ⟶ A'} {g' : B ⟶ B'} (fac : CategoryTheory.CategoryStruct.comp g f' = CategoryTheory.CategoryStruct.comp f g') :
              CommRingCat.KaehlerDifferential f ⟶ (ModuleCat.restrictScalars g'.hom).obj (CommRingCat.KaehlerDifferential f')

              The map KaehlerDifferential f ⟶ (ModuleCat.restrictScalars g').obj (KaehlerDifferential f') induced by a commutative square (given by an equality g ≫ f' = f ≫ g') in the category CommRingCat.

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                @[simp]
                theorem CommRingCat.KaehlerDifferential.map_d {A B A' B' : CommRingCat} {f : A ⟶ B} {f' : A' ⟶ B'} {g : A ⟶ A'} {g' : B ⟶ B'} (fac : CategoryTheory.CategoryStruct.comp g f' = CategoryTheory.CategoryStruct.comp f g') (b : ↑B) :
                (CommRingCat.KaehlerDifferential.map fac).hom (CommRingCat.KaehlerDifferential.d b) = CommRingCat.KaehlerDifferential.d (g'.hom b)
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                noncomputable def ModuleCat.Derivation.desc {A B : CommRingCat} {f : A ⟶ B} {M : ModuleCat ↑B} (D : M.Derivation f) :
                CommRingCat.KaehlerDifferential f ⟶ M

                Given f : A ⟶ B a morphism in the category CommRingCat, M : ModuleCat B, and D : M.Derivation f, this is the induced morphism CommRingCat.KaehlerDifferential f ⟶ M.

                Equations
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                  @[simp]
                  theorem ModuleCat.Derivation.desc_d {A B : CommRingCat} {f : A ⟶ B} {M : ModuleCat ↑B} (D : M.Derivation f) (b : ↑B) :
                  D.desc.hom (CommRingCat.KaehlerDifferential.d b) = D.d b