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[Merged by Bors] - feat: UnivLE version of EnoughInjectives in ModuleCat #8734

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[Merged by Bors] - feat: UnivLE version of EnoughInjectives in ModuleCat #8734

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feat: UnivLE version of EnoughInjectives in ModuleCat
alreadydone Nov 30, 2023
6d1c5d6
fix
alreadydone Nov 30, 2023
01dcdc7
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alreadydone Dec 4, 2023
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10 changes: 10 additions & 0 deletions Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean
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Expand Up @@ -203,6 +203,16 @@ abbrev restrictScalarsComp := restrictScalarsComp'.{v} f g _ rfl

end

noncomputable instance x {R S} [Ring R] [Ring S] (e : R ≃+* S) :
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IsEquivalence (ModuleCat.restrictScalars e.toRingHom) where
inverse := ModuleCat.restrictScalars e.symm
unitIso := NatIso.ofComponents fun M ↦ LinearEquiv.toModuleIso'
{ __ := AddEquiv.refl M
map_smul' := fun s m ↦ congr_arg (· • m) (e.right_inv s).symm }
counitIso := NatIso.ofComponents fun M ↦ LinearEquiv.toModuleIso'
{ __ := AddEquiv.refl M
map_smul' := fun r m ↦ congr_arg (· • (_ : M)) (e.left_inv r) }

open TensorProduct

variable {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] (f : R →+* S)
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8 changes: 7 additions & 1 deletion Mathlib/Algebra/Category/ModuleCat/Injective.lean
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Expand Up @@ -23,11 +23,17 @@ abelian groups.

open CategoryTheory

universe u v
universe v u
variable (R : Type u) [Ring R]

instance : EnoughInjectives (ModuleCat.{v} ℤ) :=
EnoughInjectives.of_equivalence (forget₂ (ModuleCat ℤ) AddCommGroupCat)

lemma ModuleCat.enoughInjectives : EnoughInjectives (ModuleCat.{max v u} R) :=
EnoughInjectives.of_adjunction (ModuleCat.restrictCoextendScalarsAdj.{max v u} (algebraMap ℤ R))

open ModuleCat in
instance [UnivLE.{u,v}] : EnoughInjectives (ModuleCat.{v} R) :=
letI := (equivShrink.{v} R).symm.ring
letI := enoughInjectives.{v} (Shrink.{v} R)
EnoughInjectives.of_equivalence (restrictScalars (equivShrink R).symm.ringEquiv.toRingHom)
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