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Mathlib.Algebra.Category.ModuleCat.Presheaf.ChangeOfRings

Change of presheaf of rings #

In this file, we define the restriction of scalars functor restrictScalars α : PresheafOfModules.{v} R' ⥤ PresheafOfModules.{v} R attached to a morphism of presheaves of rings α : R ⟶ R'.

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noncomputable def PresheafOfModules.restrictScalarsObj {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (M' : PresheafOfModules R') (α : R ⟶ R') :
PresheafOfModules R

The restriction of scalars of presheaves of modules, on objects.

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  • One or more equations did not get rendered due to their size.
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    @[simp]
    theorem PresheafOfModules.restrictScalarsObj_map_apply {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (M' : PresheafOfModules R') (α : R ⟶ R') {X : Cᵒᵖ} {Y : Cᵒᵖ} (f : X ⟶ Y) (a : ↑(M'.obj X)) :
    ((M'.restrictScalarsObj α).map f) a = (M'.map f) a
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    @[simp]
    theorem PresheafOfModules.restrictScalarsObj_obj {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (M' : PresheafOfModules R') (α : R ⟶ R') (X : Cᵒᵖ) :
    (M'.restrictScalarsObj α).obj X = (ModuleCat.restrictScalars (α.app X)).obj (M'.obj X)
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    noncomputable def PresheafOfModules.restrictScalars {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (α : R ⟶ R') :
    CategoryTheory.Functor (PresheafOfModules R') (PresheafOfModules R)

    The restriction of scalars functor PresheafOfModules R' ⥤ PresheafOfModules R induced by a morphism of presheaves of rings R ⟶ R'.

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      @[simp]
      theorem PresheafOfModules.restrictScalars_map_app {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (α : R ⟶ R') :
      ∀ {X Y : PresheafOfModules R'} (φ' : X ⟶ Y) (X_1 : Cᵒᵖ), ((PresheafOfModules.restrictScalars α).map φ').app X_1 = (ModuleCat.restrictScalars (α.app X_1)).map (φ'.app X_1)
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      @[simp]
      theorem PresheafOfModules.restrictScalars_obj {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (α : R ⟶ R') (M' : PresheafOfModules R') :
      (PresheafOfModules.restrictScalars α).obj M' = M'.restrictScalarsObj α
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      instance PresheafOfModules.instAdditiveRestrictScalars {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (α : R ⟶ R') :
      (PresheafOfModules.restrictScalars α).Additive
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      • ⋯ = ⋯
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      instance PresheafOfModules.instFullRestrictScalarsIdFunctorOppositeRingCat {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} :
      (PresheafOfModules.restrictScalars (CategoryTheory.CategoryStruct.id R)).Full
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      • ⋯ = ⋯
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      instance PresheafOfModules.instFaithfulRestrictScalars {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (α : R ⟶ R') :
      (PresheafOfModules.restrictScalars α).Faithful
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      • ⋯ = ⋯
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      noncomputable def PresheafOfModules.restrictScalarsCompToPresheaf {C : Type u'} [CategoryTheory.Category.{v', u'} C] {R : CategoryTheory.Functor Cᵒᵖ RingCat} {R' : CategoryTheory.Functor Cᵒᵖ RingCat} (α : R ⟶ R') :
      (PresheafOfModules.restrictScalars α).comp (PresheafOfModules.toPresheaf R) ≅ PresheafOfModules.toPresheaf R'

      The isomorphism restrictScalars α ⋙ toPresheaf R ≅ toPresheaf R' for any morphism of presheaves of rings α : R ⟶ R'.

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