feat: relative differentials as a presheaf of modules (#14014)

In this PR, we define the type `M.Derivation φ` of derivations with values in a presheaf of `R`-modules `M` relative to
a morphism of `φ : S ⟶ F.op ⋙ R` of presheaves of commutative rings over categories `C` and `D` that are related by a functor `F : C ⥤ D`. In the particular case `F` is the identity functor, we construct the universal derivation.



Co-authored-by: Joël Riou <[email protected]>

Author: joelriou

Mathlib.lean

@@ -83,6 +83,7 @@ import Mathlib.Algebra.Category.ModuleCat.Biproducts
 import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
 import Mathlib.Algebra.Category.ModuleCat.Colimits
 import Mathlib.Algebra.Category.ModuleCat.Differentials.Basic
+import Mathlib.Algebra.Category.ModuleCat.Differentials.Presheaf
 import Mathlib.Algebra.Category.ModuleCat.EpiMono
 import Mathlib.Algebra.Category.ModuleCat.FilteredColimits
 import Mathlib.Algebra.Category.ModuleCat.Free

Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean

@@ -0,0 +1,231 @@
+/-
+Copyright (c) 2024 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.Algebra.Category.ModuleCat.Presheaf
+import Mathlib.Algebra.Category.ModuleCat.Differentials.Basic
+
+/-!
+# The presheaf of differentials of a presheaf of modules
+
+In this file, we define the type `M.Derivation φ` of derivations
+with values in a presheaf of `R`-modules `M` relative to
+a morphism of `φ : S ⟶ F.op ⋙ R` of presheaves of commutative rings
+over categories `C` and `D` that are related by a functor `F : C ⥤ D`.
+We formalize the notion of universal derivation.
+
+Geometrically, if `f : X ⟶ S` is a morphisms of schemes (or more generally
+a morphism of commutative ringed spaces), we would like to apply
+these definitions in the case where `F` is the pullback functors from
+open subsets of `S` to open subsets of `X` and `φ` is the
+morphism $O_S ⟶ f_* O_X$.
+
+In order to prove that there exists a universal derivation, the target
+of which shall be called the presheaf of relative differentials of `φ`,
+we first study the case where `F` is the identity functor.
+In this case where we have a morphism of presheaves of commutative
+rings `φ' : S' ⟶ R`, we construct a derivation
+`DifferentialsConstruction.derivation'` which is universal.
+Then, the general case (TODO) shall be obtained by observing that
+derivations for `S ⟶ F.op ⋙ R` identify to derivations
+for `S' ⟶ R` where `S'` is the pullback by `F` of the presheaf of
+commutative rings `S` (the data is the same: it suffices
+to show that the two vanishing conditions `d_app` are equivalent).
+
+-/
+
+universe v u v₁ v₂ u₁ u₂
+
+open CategoryTheory
+
+variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
+
+namespace PresheafOfModules
+
+variable {S : Cᵒᵖ ⥤ CommRingCat.{u}} {F : C ⥤ D} {S' R : Dᵒᵖ ⥤ CommRingCat.{u}}
+   (M N : PresheafOfModules.{v} (R ⋙ forget₂ _ _))
+   (φ : S ⟶ F.op ⋙ R) (φ' : S' ⟶ R)
+
+/-- Given a morphism of presheaves of commutative rings `φ : S ⟶ F.op ⋙ R`,
+this is the type of relative `φ`-derivation of a presheaf of `R`-modules `M`. -/
+@[ext]
+structure Derivation where
+  /-- the underlying additive map `R.obj X →+ M.obj X` of a derivation -/
+  d {X : Dᵒᵖ} : R.obj X →+ M.obj X
+  d_mul {X : Dᵒᵖ} (a b : R.obj X) : d (a * b) = a • d b + b • d a := by aesop_cat
+  d_map {X Y : Dᵒᵖ} (f : X ⟶ Y) (x : R.obj X) :
+    d (R.map f x) = M.map f (d x) := by aesop_cat
+  d_app {X : Cᵒᵖ} (a : S.obj X) : d (φ.app X a) = 0 := by aesop_cat
+
+namespace Derivation
+
+-- Note: `d_app` cannot be a simp lemma because `dsimp` would
+-- simplify the composition of functors `R ⋙ forget₂ _ _`
+attribute [simp] d_mul d_map
+
+variable {M N φ}
+
+lemma congr_d {d d' : M.Derivation φ} (h : d = d') {X : Dᵒᵖ} (b : R.obj X) :
+    d.d b = d'.d b := by rw [h]
+
+variable (d : M.Derivation φ)
+
+@[simp] lemma d_one (X : Dᵒᵖ) : d.d (X := X) 1 = 0 := by
+  simpa using d.d_mul (X := X) 1 1
+
+/-- The postcomposition of a derivation by a morphism of presheaves of modules. -/
+@[simps! d_apply]
+def postcomp (f : M ⟶ N) : N.Derivation φ where
+  d := (f.app _).toAddMonoidHom.comp d.d
+  d_map _ _ := by simp [naturality_apply]
+  d_app {X} a := by
+    dsimp
+    erw [d_app, map_zero]
+
+/-- The universal property that a derivation `d : M.Derivation φ` must
+satisfy so that the presheaf of modules `M` can be considered as the presheaf of
+(relative) differentials of a presheaf of commutative rings `φ : S ⟶ F.op ⋙ R`. -/
+structure Universal where
+  /-- An absolyte derivation of `M'` descends as a morphism `M ⟶ M'`. -/
+  desc {M' : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)}
+    (d' : M'.Derivation φ) : M ⟶ M'
+  fac {M' : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)}
+    (d' : M'.Derivation φ) : d.postcomp (desc d') = d' := by aesop_cat
+  postcomp_injective {M' : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)}
+    (φ φ' : M ⟶ M') (h : d.postcomp φ = d.postcomp φ') : φ = φ' := by aesop_cat
+
+attribute [simp] Universal.fac
+
+instance : Subsingleton d.Universal where
+  allEq h₁ h₂ := by
+    suffices ∀ {M' : PresheafOfModules (R ⋙ forget₂ CommRingCat RingCat)}
+      (d' : M'.Derivation φ), h₁.desc d' = h₂.desc d' by
+        cases h₁
+        cases h₂
+        simp only [Universal.mk.injEq]
+        ext : 2
+        apply this
+    intro M' d'
+    apply h₁.postcomp_injective
+    simp
+
+end Derivation
+
+/-- The property that there exists a universal derivation for
+a morphism of presheaves of commutative rings `S ⟶ F.op ⋙ R`. -/
+class HasDifferentials : Prop where
+  exists_universal_derivation : ∃ (M : PresheafOfModules.{u} (R ⋙ forget₂ _ _))
+      (d : M.Derivation φ), Nonempty d.Universal
+
+/-- Given a morphism of presheaves of commutative rings `φ : S ⟶ R`,
+this is the type of relative `φ`-derivation of a presheaf of `R`-modules `M`. -/
+abbrev Derivation' : Type _ := M.Derivation (F := 𝟭 D) φ'
+
+namespace Derivation'
+
+variable {M φ'}
+
+@[simp]
+nonrec lemma d_app (d : M.Derivation' φ') {X : Dᵒᵖ} (a : S'.obj X) :
+    d.d (φ'.app X a) = 0 :=
+  d.d_app _
+
+/-- The derivation relative to the morphism of commutative rings `φ'.app X` induced by
+a derivation relative to a morphism of presheaves of commutative rings. -/
+noncomputable def app (d : M.Derivation' φ') (X : Dᵒᵖ) : (M.obj X).Derivation (φ'.app X) :=
+  ModuleCat.Derivation.mk (fun b ↦ d.d b)
+
+@[simp]
+lemma app_apply (d : M.Derivation' φ') {X : Dᵒᵖ} (b : R.obj X) :
+    (d.app X).d b = d.d b := rfl
+
+section
+
+variable (d : ∀ (X : Dᵒᵖ), (M.obj X).Derivation (φ'.app X))
+    (d_map : ∀ ⦃X Y : Dᵒᵖ⦄ (f : X ⟶ Y) (x : R.obj X),
+      (d Y).d ((R.map f) x) = (M.map f) ((d X).d x))
+
+/-- Given a morphism of presheaves of commutative rings `φ'`, this is the
+in derivation `M.Derivation' φ'` that is given by a compatible family of derivations
+with values in the modules `M.obj X` for all `X`. -/
+def mk : M.Derivation' φ' where
+  d {X} := AddMonoidHom.mk' (d X).d (by simp)
+
+@[simp]
+lemma mk_app (X : Dᵒᵖ) : (mk d d_map).app X = d X := rfl
+
+/-- Constructor for `Derivation.Universal` in the case `F` is the identity functor. -/
+def Universal.mk {d : M.Derivation' φ'}
+    (desc : ∀ {M' : PresheafOfModules (R ⋙ forget₂ _ _)}
+      (_ : M'.Derivation' φ'), M ⟶ M')
+    (fac : ∀ {M' : PresheafOfModules (R ⋙ forget₂ _ _)}
+      (d' : M'.Derivation' φ'), d.postcomp (desc d') = d')
+    (postcomp_injective : ∀ {M' : PresheafOfModules (R ⋙ forget₂ _ _)}
+      (α β : M ⟶ M'), d.postcomp α = d.postcomp β → α = β) : d.Universal where
+  desc := desc
+  fac := fac
+  postcomp_injective := postcomp_injective
+
+end
+
+end Derivation'
+
+namespace DifferentialsConstruction
+
+/-- Auxiliary definition for `relativeDifferentials'`. -/
+noncomputable def relativeDifferentials'BundledCore :
+    BundledCorePresheafOfModules.{u} (R ⋙ forget₂ _ _) where
+  obj X := CommRingCat.KaehlerDifferential (φ'.app X)
+  map f := CommRingCat.KaehlerDifferential.map (φ'.naturality f)
+
+/-- The presheaf of relative differentials of a morphism of presheaves of
+commutative rings. -/
+noncomputable def relativeDifferentials' :
+    PresheafOfModules.{u} (R ⋙ forget₂ _ _) :=
+  (relativeDifferentials'BundledCore φ').toPresheafOfModules
+
+@[simp]
+lemma relativeDifferentials'_obj (X : Dᵒᵖ) :
+    (relativeDifferentials' φ').obj X =
+      CommRingCat.KaehlerDifferential (φ'.app X) := rfl
+
+-- Note: this cannot be a simp lemma because `dsimp` would
+-- simplify the composition of functors `R ⋙ forget₂ _ _`
+lemma relativeDifferentials'_map_apply {X Y : Dᵒᵖ} (f : X ⟶ Y)
+    (x : CommRingCat.KaehlerDifferential (φ'.app X)) :
+    (relativeDifferentials' φ').map f x =
+        CommRingCat.KaehlerDifferential.map (φ'.naturality f) x := rfl
+
+lemma relativeDifferentials'_map_d {X Y : Dᵒᵖ} (f : X ⟶ Y)
+    (x : R.obj X) :
+    (relativeDifferentials' φ').map f (CommRingCat.KaehlerDifferential.d x) =
+      CommRingCat.KaehlerDifferential.d (R.map f x) := by
+  rw [relativeDifferentials'_map_apply, CommRingCat.KaehlerDifferential.map_d]
+
+/-- The universal derivation. -/
+noncomputable def derivation' : (relativeDifferentials' φ').Derivation' φ' :=
+  Derivation'.mk (fun X ↦ CommRingCat.KaehlerDifferential.D (φ'.app X)) (fun X Y f x ↦ by
+    rw [relativeDifferentials'_map_apply, CommRingCat.KaehlerDifferential.map_d])
+
+/-- The derivation `derivation' φ'` is universal. -/
+noncomputable def isUniversal' : (derivation' φ').Universal :=
+  Derivation'.Universal.mk
+    (fun {M'} d' ↦ Hom.mk'' (fun X ↦ (d'.app X).desc) (fun X Y f ↦
+      CommRingCat.KaehlerDifferential.ext (fun b ↦ by
+      dsimp [ModuleCat.ofHom]
+      erw [restrictionApp_apply, restrictionApp_apply]
+      simp only [relativeDifferentials'_map_d, ModuleCat.Derivation.desc_d,
+        d'.app_apply, d'.d_map])))
+    (fun {M'} d' ↦ by
+      ext X b
+      apply ModuleCat.Derivation.desc_d)
+    (fun {M} α β h ↦ by
+      ext1 X
+      exact CommRingCat.KaehlerDifferential.ext (Derivation.congr_d h))
+
+instance : HasDifferentials (F := 𝟭 D) φ' := ⟨_, _,  ⟨isUniversal' φ'⟩⟩
+
+end DifferentialsConstruction
+
+end PresheafOfModules

Mathlib/Algebra/Category/ModuleCat/Presheaf.lean

@@ -384,12 +384,13 @@ structure BundledCorePresheafOfModules where
   map {X Y : Cᵒᵖ} (f : X ⟶ Y) : obj X ⟶ (ModuleCat.restrictScalars (R.map f)).obj (obj Y)
   /-- `map` is compatible with the identities -/
   map_id (X : Cᵒᵖ) :
-    map (𝟙 X) = (ModuleCat.restrictScalarsId' (R.map (𝟙 X)) (R.map_id X)).inv.app (obj X)
+    map (𝟙 X) = (ModuleCat.restrictScalarsId' (R.map (𝟙 X)) (R.map_id X)).inv.app (obj X) := by
+      aesop
   /-- `map` is compatible with the composition -/
   map_comp {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
     map (f ≫ g) = map f ≫ (ModuleCat.restrictScalars (R.map f)).map (map g) ≫
       (ModuleCat.restrictScalarsComp' (R.map f) (R.map g) (R.map (f ≫ g))
-        (R.map_comp f g)).inv.app (obj Z)
+        (R.map_comp f g)).inv.app (obj Z) := by aesop
 
 namespace BundledCorePresheafOfModules