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216fb31 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | import OAI.Algebra.Finitistic.Ordinary.OrdinaryProductProjective
import Mathlib.Algebra.Homology.ShortComplex.ExactFunctor
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Limits
namespace LittleFinitistic.ProductModules
universe u
variable (L R : Type u) [Ring L] [Ring R]
instance firstFunctor_additive : (firstFunctor.{u,u} L R).Additive where
map_add := by intros; rfl
/-- Exactness on the first central-idempotent component follows by projecting a preimage. -/
theorem firstFunctor_mapsExact (K : ShortComplex (ModuleCat.{u} (L × R))) (hK : K.Exact) :
(K.map (firstFunctor L R)).Exact := by
rw [ShortComplex.moduleCat_exact_iff] at hK ⊢
intro y hy
obtain ⟨x,hx⟩ := hK y.val (congrArg Subtype.val hy)
refine ⟨⟨((1,0) : L × R) • x,?_⟩,?_⟩
· change ((1,0) : L × R) • (((1,0) : L × R) • x) = ((1,0) : L × R) • x
rw [← mul_smul]
have h : ((1,0) : L × R) * (1,0) = (1,0) := by ext <;> simp
rw [h]
· apply Subtype.ext
change K.f.hom (((1,0) : L × R) • x) = y.val
rw [K.f.hom.map_smul,hx,y.property]
instance firstFunctor_preservesHomology : (firstFunctor.{u,u} L R).PreservesHomology :=
Functor.preservesHomology_of_map_exact _ (firstFunctor_mapsExact L R)
instance firstFunctor_preservesFiniteLimits : PreservesFiniteLimits (firstFunctor.{u,u} L R) :=
Functor.preservesFiniteLimits_of_preservesHomology _
instance firstFunctor_preservesFiniteColimits : PreservesFiniteColimits (firstFunctor.{u,u} L R) :=
Functor.preservesFiniteColimits_of_preservesHomology _
/-- Taking the first component of an inflated module recovers the original module. -/
def firstInflationComponentEquiv (M : ModuleCat.{u} L) :
firstObject L R ((firstInflation L R).obj M) ≃ₗ[L] M where
toFun x := x.val
invFun x := ⟨x, one_smul L x⟩
left_inv _ := rfl
right_inv _ := rfl
map_add' _ _ := rfl
map_smul' _ _ := rfl
/-- This recovery is natural in every module and linear map. -/
def firstInflationComponentIso : firstInflation.{u,u} L R ⋙ firstFunctor L R ≅ 𝟭 (ModuleCat.{u} L) :=
NatIso.ofComponents (fun M => (firstInflationComponentEquiv L R M).toModuleIso)
(by intros; rfl)
end LittleFinitistic.ProductModules
end
end OAI
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